CINXE.COM

<!DOCTYPE html> <html lang="en" xmlns:fb="http://www.facebook.com/2008/fbml" class="wf-loading"> <head prefix="og: https://ogp.me/ns# fb: https://ogp.me/ns/fb# academia: https://ogp.me/ns/fb/academia#"> <meta charset="utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta rel="search" type="application/opensearchdescription+xml" href="/open_search.xml" title="Academia.edu"> <title>Rene Conijn - Academia.edu</title>  <link href="//a.academia-assets.com/images/favicons/favicon-production.ico" rel="shortcut icon" type="image/vnd.microsoft.icon"> <link rel="apple-touch-icon" sizes="57x57" href="//a.academia-assets.com/images/favicons/apple-touch-icon-57x57.png"> <link rel="apple-touch-icon" sizes="60x60" href="//a.academia-assets.com/images/favicons/apple-touch-icon-60x60.png"> <link rel="apple-touch-icon" sizes="72x72" href="//a.academia-assets.com/images/favicons/apple-touch-icon-72x72.png"> <link rel="apple-touch-icon" sizes="76x76" href="//a.academia-assets.com/images/favicons/apple-touch-icon-76x76.png"> <link rel="apple-touch-icon" sizes="114x114" href="//a.academia-assets.com/images/favicons/apple-touch-icon-114x114.png"> <link rel="apple-touch-icon" sizes="120x120" href="//a.academia-assets.com/images/favicons/apple-touch-icon-120x120.png"> <link rel="apple-touch-icon" sizes="144x144" href="//a.academia-assets.com/images/favicons/apple-touch-icon-144x144.png"> <link rel="apple-touch-icon" sizes="152x152" href="//a.academia-assets.com/images/favicons/apple-touch-icon-152x152.png"> <link rel="apple-touch-icon" sizes="180x180" href="//a.academia-assets.com/images/favicons/apple-touch-icon-180x180.png"> <link rel="icon" type="image/png" href="//a.academia-assets.com/images/favicons/favicon-32x32.png" sizes="32x32"> <link rel="icon" type="image/png" href="//a.academia-assets.com/images/favicons/favicon-194x194.png" sizes="194x194"> <link rel="icon" type="image/png" href="//a.academia-assets.com/images/favicons/favicon-96x96.png" sizes="96x96"> <link rel="icon" type="image/png" href="//a.academia-assets.com/images/favicons/android-chrome-192x192.png" sizes="192x192"> <link rel="icon" type="image/png" href="//a.academia-assets.com/images/favicons/favicon-16x16.png" sizes="16x16"> <link rel="manifest" href="//a.academia-assets.com/images/favicons/manifest.json"> <meta name="msapplication-TileColor" content="#2b5797"> <meta name="msapplication-TileImage" content="//a.academia-assets.com/images/favicons/mstile-144x144.png"> <meta name="theme-color" content="#ffffff"> <script> window.performance && window.performance.measure && window.performance.measure("Time To First Byte", "requestStart", "responseStart"); </script> <script> (function() { if (!window.URLSearchParams || !window.history || !window.history.replaceState) { return; } var searchParams = new URLSearchParams(window.location.search); var paramsToDelete = [ 'fs', 'sm', 'swp', 'iid', 'nbs', 'rcc', // related content category 'rcpos', // related content carousel position 'rcpg', // related carousel page 'rchid', // related content hit id 'f_ri', // research interest id, for SEO tracking 'f_fri', // featured research interest, for SEO tracking (param key without value) 'f_rid', // from research interest directory for SEO tracking 'f_loswp', // from research interest pills on LOSWP sidebar for SEO tracking 'rhid', // referrring hit id ]; if (paramsToDelete.every((key) => searchParams.get(key) === null)) { return; } paramsToDelete.forEach((key) => { searchParams.delete(key); }); var cleanUrl = new URL(window.location.href); cleanUrl.search = searchParams.toString(); history.replaceState({}, document.title, cleanUrl); })(); </script> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "profiles/works", 'action': "summary", 'controller_action': 'profiles/works#summary', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two cases. // ab_test_bucket should be of the form <ab_test_name>:<bucket> 'ab_test_bucket': null, }) </script> <script type="text/javascript"> window.sendUserTiming = function(timingName) { if (!(window.performance && window.performance.measure)) return; var entries = window.performance.getEntriesByName(timingName, "measure"); if (entries.length !== 1) return; var timingValue = Math.round(entries[0].duration); gtag('event', 'timing_complete', { name: timingName, value: timingValue, event_category: 'User-centric', }); }; window.sendUserTiming("Time To First Byte"); </script> <meta name="csrf-param" content="authenticity_token" /> <meta name="csrf-token" content="pN2/pV6y70LPxaFOh0I+9uJDTEDVn0NFoF4XjvTvC0I8XmyaQ/LdW6/55qRDHB9VQwIsLHx8QGOecTjDkp0qww==" /> <link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/wow-77f7b87cb1583fc59aa8f94756ebfe913345937eb932042b4077563bebb5fb4b.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/social/home-9e8218e1301001388038e3fc3427ed00d079a4760ff7745d1ec1b2d59103170a.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/heading-b2b823dd904da60a48fd1bfa1defd840610c2ff414d3f39ed3af46277ab8df3b.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/button-3cea6e0ad4715ed965c49bfb15dedfc632787b32ff6d8c3a474182b231146ab7.css" /><link crossorigin="" href="https://fonts.gstatic.com/" rel="preconnect" /><link href="https://fonts.googleapis.com/css2?family=DM+Sans:ital,opsz,wght@0,9..40,100..1000;1,9..40,100..1000&family=Gupter:wght@400;500;700&family=IBM+Plex+Mono:wght@300;400&family=Material+Symbols+Outlined:opsz,wght,FILL,GRAD@20,400,0,0&display=swap" rel="stylesheet" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/common-10fa40af19d25203774df2d4a03b9b5771b45109c2304968038e88a81d1215c5.css" /> <meta name="author" content="rene conijn" /> <meta name="description" content="Rene Conijn: 1 Following, 10 Research papers. Research interests: Microstrip Patch Antennas, Wavelet Transform, and Singular value decomposition." /> <meta name="google-site-verification" content="bKJMBZA7E43xhDOopFZkssMMkBRjvYERV-NaN4R6mrs" /> <script> var $controller_name = 'works'; var $action_name = "summary"; var $rails_env = 'production'; var $app_rev = 'd94ec2dec3755916737b352c28a3b5695c9eff99'; var $domain = 'academia.edu'; var $app_host = "academia.edu"; var $asset_host = "academia-assets.com"; var $start_time = new Date().getTime(); var $recaptcha_key = "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB"; var $recaptcha_invisible_key = "6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj"; var $disableClientRecordHit = false; </script> <script> window.Aedu = { hit_data: null }; window.Aedu.SiteStats = {"premium_universities_count":15276,"monthly_visitors":"113 million","monthly_visitor_count":113784677,"monthly_visitor_count_in_millions":113,"user_count":277511194,"paper_count":55203019,"paper_count_in_millions":55,"page_count":432000000,"page_count_in_millions":432,"pdf_count":16500000,"pdf_count_in_millions":16}; window.Aedu.serverRenderTime = new Date(1732771517000); window.Aedu.timeDifference = new Date().getTime() - 1732771517000; window.Aedu.isUsingCssV1 = false; window.Aedu.enableLocalization = true; window.Aedu.activateFullstory = false; window.Aedu.serviceAvailability = { status: {"attention_db":"on","bibliography_db":"on","contacts_db":"on","email_db":"on","indexability_db":"on","mentions_db":"on","news_db":"on","notifications_db":"on","offsite_mentions_db":"on","redshift":"on","redshift_exports_db":"on","related_works_db":"on","ring_db":"on","user_tests_db":"on"}, serviceEnabled: function(service) { return this.status[service] === "on"; }, readEnabled: function(service) { return this.serviceEnabled(service) || this.status[service] === "read_only"; }, }; window.Aedu.viewApmTrace = function() { // Check if x-apm-trace-id meta tag is set, and open the trace in APM // in a new window if it is. var apmTraceId = document.head.querySelector('meta[name="x-apm-trace-id"]'); if (apmTraceId) { var traceId = apmTraceId.content; // Use trace ID to construct URL, an example URL looks like: // https://app.datadoghq.com/apm/traces?query=trace_id%31298410148923562634 var apmUrl = 'https://app.datadoghq.com/apm/traces?query=trace_id%3A' + traceId; window.open(apmUrl, '_blank'); } }; </script>  <link href="https://fonts.googleapis.com/css?family=Roboto:100,100i,300,300i,400,400i,500,500i,700,700i,900,900i" rel="stylesheet"> <link href="//maxcdn.bootstrapcdn.com/font-awesome/4.3.0/css/font-awesome.min.css" rel="stylesheet"> <link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/libraries-a9675dcb01ec4ef6aa807ba772c7a5a00c1820d3ff661c1038a20f80d06bb4e4.css" /> <link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/academia-bdb9e8c097f01e611f2fc5e2f1a9dc599beede975e2ae5629983543a1726e947.css" /> <link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system_legacy-056a9113b9a0f5343d013b29ee1929d5a18be35fdcdceb616600b4db8bd20054.css" /> <script src="//a.academia-assets.com/assets/webpack_bundles/runtime-bundle-005434038af4252ca37c527588411a3d6a0eabb5f727fac83f8bbe7fd88d93bb.js"></script> <script src="//a.academia-assets.com/assets/webpack_bundles/webpack_libraries_and_infrequently_changed.wjs-bundle-bae13f9b51961d5f1e06008e39e31d0138cb31332e8c2e874c6d6a250ec2bb14.js"></script> <script src="//a.academia-assets.com/assets/webpack_bundles/core_webpack.wjs-bundle-19a25d160d01bde427443d06cd6b810c4c92c6026e7cb31519e06313eb24ed90.js"></script> <script src="//a.academia-assets.com/assets/webpack_bundles/sentry.wjs-bundle-5fe03fddca915c8ba0f7edbe64c194308e8ce5abaed7bffe1255ff37549c4808.js"></script> <script> jade = window.jade || {}; jade.helpers = window.$h; jade._ = window._; </script>  <script id="tag-manager-head-root">(function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= 'https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); })(window,document,'script','dataLayer_old','GTM-5G9JF7Z');</script>  <script> window.gptadslots = []; window.googletag = window.googletag || {}; window.googletag.cmd = window.googletag.cmd || []; </script> <script type="text/javascript"> // TODO(jacob): This should be defined, may be rare load order problem. // Checking if null is just a quick fix, will default to en if unset. // Better fix is to run this immedietely after I18n is set. if (window.I18n != null) { I18n.defaultLocale = "en"; I18n.locale = "en"; I18n.fallbacks = true; } </script> <link rel="canonical" href="https://independent.academia.edu/ReneConijn" /> </head>   <body class='c-profiles/works a-summary logged_out'>  <div id="fb-root"></div><script>window.fbAsyncInit = function() { FB.init({ appId: "2369844204", version: "v8.0", status: true, cookie: true, xfbml: true }); // Additional initialization code. if (window.InitFacebook) { // facebook.ts already loaded, set it up. window.InitFacebook(); } else { // Set a flag for facebook.ts to find when it loads. window.academiaAuthReadyFacebook = true; } };</script><script>window.fbAsyncLoad = function() { // Protection against double calling of this function if (window.FB) { return; } (function(d, s, id){ var js, fjs = d.getElementsByTagName(s)[0]; if (d.getElementById(id)) {return;} js = d.createElement(s); js.id = id; js.src = "//connect.facebook.net/en_US/sdk.js"; fjs.parentNode.insertBefore(js, fjs); }(document, 'script', 'facebook-jssdk')); } if (!window.defer_facebook) { // Autoload if not deferred window.fbAsyncLoad(); } else { // Defer loading by 5 seconds setTimeout(function() { window.fbAsyncLoad(); }, 5000); }</script> <div id="google-root"></div><script>window.loadGoogle = function() { if (window.InitGoogle) { // google.ts already loaded, set it up. window.InitGoogle("331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b"); } else { // Set a flag for google.ts to use when it loads. window.GoogleClientID = "331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b"; } };</script><script>window.googleAsyncLoad = function() { // Protection against double calling of this function (function(d) { var js; var id = 'google-jssdk'; var ref = d.getElementsByTagName('script')[0]; if (d.getElementById(id)) { return; } js = d.createElement('script'); js.id = id; js.async = true; js.onload = loadGoogle; js.src = "https://accounts.google.com/gsi/client" ref.parentNode.insertBefore(js, ref); }(document)); } if (!window.defer_google) { // Autoload if not deferred window.googleAsyncLoad(); } else { // Defer loading by 5 seconds setTimeout(function() { window.googleAsyncLoad(); }, 5000); }</script> <div id="tag-manager-body-root">  <noscript><iframe src="https://www.googletagmanager.com/ns.html?id=GTM-5G9JF7Z" height="0" width="0" style="display:none;visibility:hidden"></iframe></noscript>   <script> window.addEventListener('load', function() { if (document.querySelector('input[name="commit"]')) { document.querySelector('input[name="commit"]').addEventListener('click', function() { gtag('event', 'click', { event_category: 'button', event_label: 'Log In' }) }) } }); </script> </div> <script>var _comscore = _comscore || []; _comscore.push({ c1: "2", c2: "26766707" }); (function() { var s = document.createElement("script"), el = document.getElementsByTagName("script")[0]; s.async = true; s.src = (document.location.protocol == "https:" ? "https://sb" : "http://b") + ".scorecardresearch.com/beacon.js"; el.parentNode.insertBefore(s, el); })();</script><img src="https://sb.scorecardresearch.com/p?c1=2&c2=26766707&cv=2.0&cj=1" style="position: absolute; visibility: hidden" /> <div id='react-modal'></div> <div class='DesignSystem'> <a class='u-showOnFocus' href='#site'> Skip to main content </a> </div> <div id="upgrade_ie_banner" style="display: none;"><p>Academia.edu no longer supports Internet Explorer.</p><p>To browse Academia.edu and the wider internet faster and more securely, please take a few seconds to <a href="https://www.academia.edu/upgrade-browser">upgrade your browser</a>.</p></div><script>// Show this banner for all versions of IE if (!!window.MSInputMethodContext || /(MSIE)/.test(navigator.userAgent)) { document.getElementById('upgrade_ie_banner').style.display = 'block'; }</script> <div class="DesignSystem bootstrap ShrinkableNav"><div class="navbar navbar-default main-header"><div class="container-wrapper" id="main-header-container"><div class="container"><div class="navbar-header"><div class="nav-left-wrapper u-mt0x"><div class="nav-logo"><a data-main-header-link-target="logo_home" href="https://www.academia.edu/"><img class="visible-xs-inline-block" style="height: 24px;" alt="Academia.edu" src="//a.academia-assets.com/images/academia-logo-redesign-2015-A.svg" width="24" height="24" /><img width="145.2" height="18" class="hidden-xs" style="height: 24px;" alt="Academia.edu" src="//a.academia-assets.com/images/academia-logo-redesign-2015.svg" /></a></div><div class="nav-search"><div class="SiteSearch-wrapper select2-no-default-pills"><form class="js-SiteSearch-form DesignSystem" action="https://www.academia.edu/search" accept-charset="UTF-8" method="get"><input name="utf8" type="hidden" value="✓" autocomplete="off" /><i class="SiteSearch-icon fa fa-search u-fw700 u-positionAbsolute u-tcGrayDark"></i><input class="js-SiteSearch-form-input SiteSearch-form-input form-control" data-main-header-click-target="search_input" name="q" placeholder="Search" type="text" value="" /></form></div></div></div><div class="nav-right-wrapper pull-right"><ul class="NavLinks js-main-nav list-unstyled"><li class="NavLinks-link"><a class="js-header-login-url Button Button--inverseGray Button--sm u-mb4x" id="nav_log_in" rel="nofollow" href="https://www.academia.edu/login">Log In</a></li><li class="NavLinks-link u-p0x"><a class="Button Button--inverseGray Button--sm u-mb4x" rel="nofollow" href="https://www.academia.edu/signup">Sign Up</a></li></ul><button class="hidden-lg hidden-md hidden-sm u-ml4x navbar-toggle collapsed" data-target=".js-mobile-header-links" data-toggle="collapse" type="button"><span class="icon-bar"></span><span class="icon-bar"></span><span class="icon-bar"></span></button></div></div><div class="collapse navbar-collapse js-mobile-header-links"><ul class="nav navbar-nav"><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://www.academia.edu/login">Log In</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://www.academia.edu/signup">Sign Up</a></li><li class="u-borderColorGrayLight u-borderBottom1 js-mobile-nav-expand-trigger"><a href="#">more&nbsp<span class="caret"></span></a></li><li><ul class="js-mobile-nav-expand-section nav navbar-nav u-m0x collapse"><li class="u-borderColorGrayLight u-borderBottom1"><a rel="false" href="https://www.academia.edu/about">About</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://www.academia.edu/press">Press</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://medium.com/@academia">Blog</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="false" href="https://www.academia.edu/documents">Papers</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://www.academia.edu/terms">Terms</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://www.academia.edu/privacy">Privacy</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://www.academia.edu/copyright">Copyright</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://www.academia.edu/hiring"><i class="fa fa-briefcase"></i> We're Hiring!</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://support.academia.edu/"><i class="fa fa-question-circle"></i> Help Center</a></li><li class="js-mobile-nav-collapse-trigger u-borderColorGrayLight u-borderBottom1 dropup" style="display:none"><a href="#">less&nbsp<span class="caret"></span></a></li></ul></li></ul></div></div></div><script>(function(){ var $moreLink = $(".js-mobile-nav-expand-trigger"); var $lessLink = $(".js-mobile-nav-collapse-trigger"); var $section = $('.js-mobile-nav-expand-section'); $moreLink.click(function(ev){ ev.preventDefault(); $moreLink.hide(); $lessLink.show(); $section.collapse('show'); }); $lessLink.click(function(ev){ ev.preventDefault(); $moreLink.show(); $lessLink.hide(); $section.collapse('hide'); }); })() if ($a.is_logged_in() || false) { new Aedu.NavigationController({ el: '.js-main-nav', showHighlightedNotification: false }); } else { $(".js-header-login-url").attr("href", $a.loginUrlWithRedirect()); } Aedu.autocompleteSearch = new AutocompleteSearch({el: '.js-SiteSearch-form'});</script></div></div> <div id='site' class='fixed'> <div id="content" class="clearfix"> <script>document.addEventListener('DOMContentLoaded', function(){ var $dismissible = $(".dismissible_banner"); $dismissible.click(function(ev) { $dismissible.hide(); }); });</script> <script src="//a.academia-assets.com/assets/webpack_bundles/profile.wjs-bundle-e032f1d55548c2f2dee4eac9fe52f38beaf13471f2298bb2ea82725ae930b83c.js" defer="defer"></script><script>Aedu.rankings = { showPaperRankingsLink: false } $viewedUser = Aedu.User.set_viewed( {"id":102711148,"first_name":"Rene","middle_initials":null,"last_name":"Conijn","page_name":"ReneConijn","domain_name":"independent","created_at":"2019-02-18T02:03:26.325-08:00","display_name":"Rene Conijn","url":"https://independent.academia.edu/ReneConijn","photo":"/images/s65_no_pic.png","has_photo":false,"is_analytics_public":false,"interests":[{"id":369147,"name":"Microstrip Patch Antennas","url":"https://www.academia.edu/Documents/in/Microstrip_Patch_Antennas"},{"id":91262,"name":"Wavelet Transform","url":"https://www.academia.edu/Documents/in/Wavelet_Transform"},{"id":85880,"name":"Singular value decomposition","url":"https://www.academia.edu/Documents/in/Singular_value_decomposition"},{"id":123396,"name":"Sound and Vibration","url":"https://www.academia.edu/Documents/in/Sound_and_Vibration"},{"id":1687256,"name":"Electrical Engineering and Computer Science","url":"https://www.academia.edu/Documents/in/Electrical_Engineering_and_Computer_Science"}]} ); if ($a.is_logged_in() && $viewedUser.is_current_user()) { $('body').addClass('profile-viewed-by-owner'); } $socialProfiles = []</script><div id="js-react-on-rails-context" style="display:none" data-rails-context="{"inMailer":false,"i18nLocale":"en","i18nDefaultLocale":"en","href":"https://independent.academia.edu/ReneConijn","location":"/ReneConijn","scheme":"https","host":"independent.academia.edu","port":null,"pathname":"/ReneConijn","search":null,"httpAcceptLanguage":null,"serverSide":false}"></div> <div class="js-react-on-rails-component" style="display:none" data-component-name="ProfileCheckPaperUpdate" data-props="{}" data-trace="false" data-dom-id="ProfileCheckPaperUpdate-react-component-e4418de1-a0fb-4833-8678-665c095b9c12"></div> <div id="ProfileCheckPaperUpdate-react-component-e4418de1-a0fb-4833-8678-665c095b9c12"></div> <div class="DesignSystem"><div class="onsite-ping" id="onsite-ping"></div></div><div class="profile-user-info DesignSystem"><div class="social-profile-container"><div class="left-panel-container"><div class="user-info-component-wrapper"><div class="user-summary-cta-container"><div class="user-summary-container"><div class="social-profile-avatar-container"><img class="profile-avatar u-positionAbsolute" border="0" alt="" src="//a.academia-assets.com/images/s200_no_pic.png" /></div><div class="title-container"><h1 class="ds2-5-heading-sans-serif-sm">Rene Conijn</h1><div class="affiliations-container fake-truncate js-profile-affiliations"></div></div></div><div class="sidebar-cta-container"><button class="ds2-5-button hidden profile-cta-button grow js-profile-follow-button" data-broccoli-component="user-info.follow-button" data-click-track="profile-user-info-follow-button" data-follow-user-fname="Rene" data-follow-user-id="102711148" data-follow-user-source="profile_button" data-has-google="false"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">add</span>Follow</button><button class="ds2-5-button hidden profile-cta-button grow js-profile-unfollow-button" data-broccoli-component="user-info.unfollow-button" data-click-track="profile-user-info-unfollow-button" data-unfollow-user-id="102711148"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">done</span>Following</button></div></div><div class="user-stats-container"><a><div class="stat-container js-profile-followers"><p class="label">Followers</p><p class="data">0</p></div></a><a><div class="stat-container js-profile-followees" data-broccoli-component="user-info.followees-count" data-click-track="profile-expand-user-info-following"><p class="label">Following</p><p class="data">1</p></div></a><span><div class="stat-container"><p class="label"><span class="js-profile-total-view-text">Public Views</span></p><p class="data"><span class="js-profile-view-count"></span></p></div></span></div><div class="ri-section"><div class="ri-section-header"><span>Interests</span></div><div class="ri-tags-container"><a data-click-track="profile-user-info-expand-research-interests" data-has-card-for-ri-list="102711148" href="https://www.academia.edu/Documents/in/Microstrip_Patch_Antennas"><div id="js-react-on-rails-context" style="display:none" data-rails-context="{"inMailer":false,"i18nLocale":"en","i18nDefaultLocale":"en","href":"https://independent.academia.edu/ReneConijn","location":"/ReneConijn","scheme":"https","host":"independent.academia.edu","port":null,"pathname":"/ReneConijn","search":null,"httpAcceptLanguage":null,"serverSide":false}"></div> <div class="js-react-on-rails-component" style="display:none" data-component-name="Pill" data-props="{"color":"gray","children":["Microstrip Patch Antennas"]}" data-trace="false" data-dom-id="Pill-react-component-7a3b3a0f-00f0-4ff2-9f09-52502cc40009"></div> <div id="Pill-react-component-7a3b3a0f-00f0-4ff2-9f09-52502cc40009"></div> </a><a data-click-track="profile-user-info-expand-research-interests" data-has-card-for-ri-list="102711148" href="https://www.academia.edu/Documents/in/Wavelet_Transform"><div class="js-react-on-rails-component" style="display:none" data-component-name="Pill" data-props="{"color":"gray","children":["Wavelet Transform"]}" data-trace="false" data-dom-id="Pill-react-component-2ced03af-9f6f-4b42-8fc8-d7740147af0b"></div> <div id="Pill-react-component-2ced03af-9f6f-4b42-8fc8-d7740147af0b"></div> </a><a data-click-track="profile-user-info-expand-research-interests" data-has-card-for-ri-list="102711148" href="https://www.academia.edu/Documents/in/Singular_value_decomposition"><div class="js-react-on-rails-component" style="display:none" data-component-name="Pill" data-props="{"color":"gray","children":["Singular value decomposition"]}" data-trace="false" data-dom-id="Pill-react-component-99b866cd-a438-4c0d-b761-9e074484befc"></div> <div id="Pill-react-component-99b866cd-a438-4c0d-b761-9e074484befc"></div> </a><a data-click-track="profile-user-info-expand-research-interests" data-has-card-for-ri-list="102711148" href="https://www.academia.edu/Documents/in/Sound_and_Vibration"><div class="js-react-on-rails-component" style="display:none" data-component-name="Pill" data-props="{"color":"gray","children":["Sound and Vibration"]}" data-trace="false" data-dom-id="Pill-react-component-67d678e0-8d8e-4956-a8ef-eaa4eff38d88"></div> <div id="Pill-react-component-67d678e0-8d8e-4956-a8ef-eaa4eff38d88"></div> </a><a data-click-track="profile-user-info-expand-research-interests" data-has-card-for-ri-list="102711148" href="https://www.academia.edu/Documents/in/Electrical_Engineering_and_Computer_Science"><div class="js-react-on-rails-component" style="display:none" data-component-name="Pill" data-props="{"color":"gray","children":["Electrical Engineering and Computer Science"]}" data-trace="false" data-dom-id="Pill-react-component-da755f0c-627a-4256-b804-73ba44dfc730"></div> <div id="Pill-react-component-da755f0c-627a-4256-b804-73ba44dfc730"></div> </a></div></div></div></div><div class="right-panel-container"><div class="user-content-wrapper"><div class="uploads-container" id="social-redesign-work-container"><div class="upload-header"><h2 class="ds2-5-heading-sans-serif-xs">Uploads</h2></div><div class="documents-container backbone-social-profile-documents" style="width: 100%;"><div class="u-taCenter"></div><div class="profile--tab_content_container js-tab-pane tab-pane active" id="all"><div class="profile--tab_heading_container js-section-heading" data-section="Papers" id="Papers"><h3 class="profile--tab_heading_container">Papers by Rene Conijn</h3></div><div class="js-work-strip profile--work_container" data-work-id="84336774"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/84336774/On_the_size_of_the_largest_cluster_in_2D_critical_percolation"><img alt="Research paper thumbnail of On the size of the largest cluster in 2D critical percolation" class="work-thumbnail" src="https://attachments.academia-assets.com/89395086/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/84336774/On_the_size_of_the_largest_cluster_in_2D_critical_percolation">On the size of the largest cluster in 2D critical percolation</a></div><div class="wp-workCard_item"><span>arXiv: Probability</span><span>, Aug 20, 2012</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="c103fee90bcdf8985596c823af0663c5" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":89395086,"asset_id":84336774,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/89395086/download_file?st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="84336774"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="84336774"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 84336774; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=84336774]").text(description); $(".js-view-count[data-work-id=84336774]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 84336774; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='84336774']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 84336774, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "c103fee90bcdf8985596c823af0663c5" } } $('.js-work-strip[data-work-id=84336774]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":84336774,"title":"On the size of the largest cluster in 2D critical percolation","translated_title":"","metadata":{"grobid_abstract":"We consider (near-)critical percolation on the square lattice. Let Mn be the size of the largest open cluster contained in the box [−n, n] 2 , and let π(n) be the probability that there is an open path from O to the boundary of the box. It is well-known (see [BCKS01]) that for all 0 \u003c a \u003c b the probability that Mn is smaller than an 2 π(n) and the probability that Mn is larger than bn 2 π(n) are bounded away from 0 as n → ∞. It is a natural question, which arises for instance in the study of so-called frozenpercolation processes, if a similar result holds for the probability that Mn is between an 2 π(n) and bn 2 π(n). By a suitable partition of the box, and a careful construction involving the building blocks, we show that the answer to this question is affirmative. The 'sublinearity' of 1/π(n) appears to be essential for the argument.","publication_date":{"day":20,"month":8,"year":2012,"errors":{}},"publication_name":"arXiv: Probability","grobid_abstract_attachment_id":89395086},"translated_abstract":null,"internal_url":"https://www.academia.edu/84336774/On_the_size_of_the_largest_cluster_in_2D_critical_percolation","translated_internal_url":"","created_at":"2022-08-08T09:15:41.838-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":102711148,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":89395086,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/89395086/thumbnails/1.jpg","file_name":"1208.4014.pdf","download_url":"https://www.academia.edu/attachments/89395086/download_file?st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_the_size_of_the_largest_cluster_in_2D.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/89395086/1208.4014-libre.pdf?1659976532=\u0026response-content-disposition=attachment%3B+filename%3DOn_the_size_of_the_largest_cluster_in_2D.pdf\u0026Expires=1732775116\u0026Signature=YBrp6VV5dQET1cLUObUJQfKpLrtYsES26mrJXgz~oRIe~RRFb9bwjDL6-wbaIQA-SsTjkfXbBWSIcqjeRbOTW5-J3PHA0So4lHrAw5gIaTfM5sOsdDXbWAeJABLQYauA2iJZ9wknj5Yotak1-c-HK4QjNjTRGE337OYLC25gNescEHSKLj8z~mnbmDn-iEXYsTsYfKE5gS9jFzB~S1gA1JtfvN14EmQ8jUuSe3GpEZDo9T1LUgFzv2Y16uRYTWjxm3CLwPrirrM41C0Dj6zLpLHioVEifmqWbt8fj8ByGbnRGPdGTDxW0CEX5IFaJgYWqzLjRy0GggT0kTYeB311dw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"On_the_size_of_the_largest_cluster_in_2D_critical_percolation","translated_slug":"","page_count":12,"language":"en","content_type":"Work","owner":{"id":102711148,"first_name":"Rene","middle_initials":null,"last_name":"Conijn","page_name":"ReneConijn","domain_name":"independent","created_at":"2019-02-18T02:03:26.325-08:00","display_name":"Rene Conijn","url":"https://independent.academia.edu/ReneConijn"},"attachments":[{"id":89395086,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/89395086/thumbnails/1.jpg","file_name":"1208.4014.pdf","download_url":"https://www.academia.edu/attachments/89395086/download_file?st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_the_size_of_the_largest_cluster_in_2D.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/89395086/1208.4014-libre.pdf?1659976532=\u0026response-content-disposition=attachment%3B+filename%3DOn_the_size_of_the_largest_cluster_in_2D.pdf\u0026Expires=1732775116\u0026Signature=YBrp6VV5dQET1cLUObUJQfKpLrtYsES26mrJXgz~oRIe~RRFb9bwjDL6-wbaIQA-SsTjkfXbBWSIcqjeRbOTW5-J3PHA0So4lHrAw5gIaTfM5sOsdDXbWAeJABLQYauA2iJZ9wknj5Yotak1-c-HK4QjNjTRGE337OYLC25gNescEHSKLj8z~mnbmDn-iEXYsTsYfKE5gS9jFzB~S1gA1JtfvN14EmQ8jUuSe3GpEZDo9T1LUgFzv2Y16uRYTWjxm3CLwPrirrM41C0Dj6zLpLHioVEifmqWbt8fj8ByGbnRGPdGTDxW0CEX5IFaJgYWqzLjRy0GggT0kTYeB311dw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":892,"name":"Statistics","url":"https://www.academia.edu/Documents/in/Statistics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":385916,"name":"Percolation threshold","url":"https://www.academia.edu/Documents/in/Percolation_threshold"}],"urls":[{"id":22726228,"url":"https://arxiv.org/pdf/1208.4014.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="72797302"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/72797302/The_gaps_between_the_sizes_of_large_clusters_in_2D_critical_percolation"><img alt="Research paper thumbnail of The gaps between the sizes of large clusters in 2D critical percolation" class="work-thumbnail" src="https://attachments.academia-assets.com/81581237/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/72797302/The_gaps_between_the_sizes_of_large_clusters_in_2D_critical_percolation">The gaps between the sizes of large clusters in 2D critical percolation</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Consider critical bond percolation on a large 2n by 2n box on the square lattice. It is well-know...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Consider critical bond percolation on a large 2n by 2n box on the square lattice. It is well-known that the size (i.e. number of vertices) of the largest open cluster is, with high probability, of order n^2 π(n), where π(n) denotes the probability that there is an open path from the center to the boundary of the box. The same result holds for the second-largest cluster, the third largest cluster etcetera. Jarai showed that the differences between the sizes of these clusters is, with high probability, at least of order √(n^2 π(n)). Although this bound was enough for his applications (to incipient infinite clusters), he believed, but had no proof, that the differences are in fact of the same order as the cluster sizes themselves, i.e. n^2 π(n). Our main result is a proof that this is indeed the case.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="633516dcc8b5ae3043acbfb4da14573f" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":81581237,"asset_id":72797302,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/81581237/download_file?st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="72797302"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="72797302"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 72797302; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=72797302]").text(description); $(".js-view-count[data-work-id=72797302]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 72797302; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='72797302']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 72797302, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "633516dcc8b5ae3043acbfb4da14573f" } } $('.js-work-strip[data-work-id=72797302]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":72797302,"title":"The gaps between the sizes of large clusters in 2D critical percolation","translated_title":"","metadata":{"abstract":"Consider critical bond percolation on a large 2n by 2n box on the square lattice. It is well-known that the size (i.e. number of vertices) of the largest open cluster is, with high probability, of order n^2 π(n), where π(n) denotes the probability that there is an open path from the center to the boundary of the box. The same result holds for the second-largest cluster, the third largest cluster etcetera. Jarai showed that the differences between the sizes of these clusters is, with high probability, at least of order √(n^2 π(n)). Although this bound was enough for his applications (to incipient infinite clusters), he believed, but had no proof, that the differences are in fact of the same order as the cluster sizes themselves, i.e. n^2 π(n). Our main result is a proof that this is indeed the case.","publication_date":{"day":8,"month":10,"year":2013,"errors":{}}},"translated_abstract":"Consider critical bond percolation on a large 2n by 2n box on the square lattice. It is well-known that the size (i.e. number of vertices) of the largest open cluster is, with high probability, of order n^2 π(n), where π(n) denotes the probability that there is an open path from the center to the boundary of the box. The same result holds for the second-largest cluster, the third largest cluster etcetera. Jarai showed that the differences between the sizes of these clusters is, with high probability, at least of order √(n^2 π(n)). Although this bound was enough for his applications (to incipient infinite clusters), he believed, but had no proof, that the differences are in fact of the same order as the cluster sizes themselves, i.e. n^2 π(n). Our main result is a proof that this is indeed the case.","internal_url":"https://www.academia.edu/72797302/The_gaps_between_the_sizes_of_large_clusters_in_2D_critical_percolation","translated_internal_url":"","created_at":"2022-03-02T06:52:58.727-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":102711148,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":81581237,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/81581237/thumbnails/1.jpg","file_name":"1310.2019v1.pdf","download_url":"https://www.academia.edu/attachments/81581237/download_file?st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_gaps_between_the_sizes_of_large_clus.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/81581237/1310.2019v1-libre.pdf?1646233506=\u0026response-content-disposition=attachment%3B+filename%3DThe_gaps_between_the_sizes_of_large_clus.pdf\u0026Expires=1732775116\u0026Signature=TAsVQBKrhNzHiJAI0FYjXEfnB1cP8hByQJD59oyeOfA5gpvJCHbCO3smAS6Q6p9k9FVokQlF1pnRueO4MWnwQD4gHf6tCc3nxNHH5u7E-rmY7EV3P6MRI~5kAc~OzB3KiNoIyrxK0ZZd5pxvmtE-0wPYq84CDYMzTbRRxfKJcQTSTk~e0808v2-VnC3Emn1B-vKXVCl9Zs8vO5dVEzu3RRyEMSUqB-31tJTAuuwhSSxD8JxPdWcTdXzJc9wYPDLU8tkA1gvrlJ~jxkxxR~xkKqWaY8sQz-ecmNmWbi-0U1WfYYoXEvrLzX1QadmtT9efEWRQUPMMnkwk9SfIlHOcOQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"The_gaps_between_the_sizes_of_large_clusters_in_2D_critical_percolation","translated_slug":"","page_count":10,"language":"en","content_type":"Work","owner":{"id":102711148,"first_name":"Rene","middle_initials":null,"last_name":"Conijn","page_name":"ReneConijn","domain_name":"independent","created_at":"2019-02-18T02:03:26.325-08:00","display_name":"Rene Conijn","url":"https://independent.academia.edu/ReneConijn"},"attachments":[{"id":81581237,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/81581237/thumbnails/1.jpg","file_name":"1310.2019v1.pdf","download_url":"https://www.academia.edu/attachments/81581237/download_file?st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_gaps_between_the_sizes_of_large_clus.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/81581237/1310.2019v1-libre.pdf?1646233506=\u0026response-content-disposition=attachment%3B+filename%3DThe_gaps_between_the_sizes_of_large_clus.pdf\u0026Expires=1732775116\u0026Signature=TAsVQBKrhNzHiJAI0FYjXEfnB1cP8hByQJD59oyeOfA5gpvJCHbCO3smAS6Q6p9k9FVokQlF1pnRueO4MWnwQD4gHf6tCc3nxNHH5u7E-rmY7EV3P6MRI~5kAc~OzB3KiNoIyrxK0ZZd5pxvmtE-0wPYq84CDYMzTbRRxfKJcQTSTk~e0808v2-VnC3Emn1B-vKXVCl9Zs8vO5dVEzu3RRyEMSUqB-31tJTAuuwhSSxD8JxPdWcTdXzJc9wYPDLU8tkA1gvrlJ~jxkxxR~xkKqWaY8sQz-ecmNmWbi-0U1WfYYoXEvrLzX1QadmtT9efEWRQUPMMnkwk9SfIlHOcOQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":81581235,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/81581235/thumbnails/1.jpg","file_name":"1310.2019v1.pdf","download_url":"https://www.academia.edu/attachments/81581235/download_file","bulk_download_file_name":"The_gaps_between_the_sizes_of_large_clus.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/81581235/1310.2019v1-libre.pdf?1646233507=\u0026response-content-disposition=attachment%3B+filename%3DThe_gaps_between_the_sizes_of_large_clus.pdf\u0026Expires=1732775116\u0026Signature=GaZe-KJAi1cgi-N~4RPgJKr2IKkiYDsuKgSTMmmzo1BQtLv4SjnYpIcE2OmMuFKhhrXTsOuJgqOM0dbcLSKn~~jZ13-DXCE~HEj7vbug5-Y4~ww7UklfwDtozlBFuM8ACAZJuXU2vrJEnXYw-4Y3uTgOwTBzZGpF4hAgMcPgX2cgmG2skAyMxlyZ8Ip7hl4fEVXXnqXXScpZtP20H3D8KEZCmSRK6QTYCa8AtzmfvHBOCVX9B-Sm8eRCbfspYJHjmx34We9Yw0uoghPNBekjdGuJFZUrAuBe~NvM-dXHXe-TvyMezIkU574Il6X7vjIJ7jt06feLas0Nb5sywn5OAg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":892,"name":"Statistics","url":"https://www.academia.edu/Documents/in/Statistics"}],"urls":[{"id":18152043,"url":"https://arxiv.org/pdf/1310.2019v1.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="72797299"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/72797299/Planar_Critical_Percolation_Large_clusters_and_Scaling_limits"><img alt="Research paper thumbnail of Planar Critical Percolation: Large clusters and Scaling limits" class="work-thumbnail" src="https://attachments.academia-assets.com/81581233/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/72797299/Planar_Critical_Percolation_Large_clusters_and_Scaling_limits">Planar Critical Percolation: Large clusters and Scaling limits</a></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="69691eecfe3c4266d0c53ce23d04e541" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":81581233,"asset_id":72797299,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/81581233/download_file?st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="72797299"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="72797299"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 72797299; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=72797299]").text(description); $(".js-view-count[data-work-id=72797299]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 72797299; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='72797299']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 72797299, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "69691eecfe3c4266d0c53ce23d04e541" } } $('.js-work-strip[data-work-id=72797299]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":72797299,"title":"Planar Critical Percolation: Large clusters and Scaling limits","translated_title":"","metadata":{"grobid_abstract":"This chapter is based on [10] with Rob van den Berg. We consider (near-)critical percolation on the square lattice. Let M n be the size of the largest open cluster contained in the box [−n, n] 2 , and let π(n) be the probability that there is an open path from O to the boundary of the box. It is well-known (see [17]) that for all 0 \u003c a \u003c b the probability that M n is smaller than an 2 π(n) and the probability that M n is larger than bn 2 π(n) are bounded away from 0 as n → ∞. It is a natural question, which arises for instance in the study of so-called frozenpercolation processes, if a similar result holds for the probability that M n is between an 2 π(n) and bn 2 π(n). By a suitable partition of the box, and a careful construction involving the building blocks, we show that the answer to this question is armative. The`sublinearity' of 1/π(n) appears to be essential for the argument. percolation and FK-Ising This chapter is based on [20] with Federico Camia and Demeter Kiss. Under some general assumptions we construct the scaling limit of open clusters and their associated counting measures in a class of two-dimensional percolation models. Our results apply, in particular, to critical Bernoulli site percolation on the triangular lattice. We also provide conditional results for the critical FK-Ising model on the square lattice. Fundamental properties of the scaling limit, such as conformal covariance, are explored. Applications such as the scaling limit of the largest cluster in a bounded domain and a geometric representation of the magnetization eld for the critical Ising model are presented.","publication_date":{"day":null,"month":null,"year":2015,"errors":{}},"grobid_abstract_attachment_id":81581233},"translated_abstract":null,"internal_url":"https://www.academia.edu/72797299/Planar_Critical_Percolation_Large_clusters_and_Scaling_limits","translated_internal_url":"","created_at":"2022-03-02T06:52:57.953-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":102711148,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":81581233,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/81581233/thumbnails/1.jpg","file_name":"23595B.pdf","download_url":"https://www.academia.edu/attachments/81581233/download_file?st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Planar_Critical_Percolation_Large_cluste.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/81581233/23595B-libre.pdf?1646233518=\u0026response-content-disposition=attachment%3B+filename%3DPlanar_Critical_Percolation_Large_cluste.pdf\u0026Expires=1732775116\u0026Signature=EOxAA3P5GtO5WLKQvDVz2pf5u0KFuihA3mO3VHk2Q73ia2Lk1SpevxMLgE55Ymy~GHClnsQ5rs6sRElv8YKmdGwY~jfgUvZHkoFR6ryz7ViJbmJ9xWi3S4UhN7Ua9OHXVGW8JOx-~R384pCDeIAK6zOgOwxArvUWNmpUfKObdWId8UALUt0-L-T-f58iftj97nnUCMFlAl9zVkr39uME~f-DdJEpSBaNrm1DCeanZq9AP3booeOavV8G72PM0mRG4PD3bkVX1zABLaruEUGuQ-icqvc82gF-vQJjEeDjOwKsDzcA-nSczx32QGwKd6bKNN5JEFXxCowlUqSJ552Rqg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Planar_Critical_Percolation_Large_clusters_and_Scaling_limits","translated_slug":"","page_count":118,"language":"en","content_type":"Work","owner":{"id":102711148,"first_name":"Rene","middle_initials":null,"last_name":"Conijn","page_name":"ReneConijn","domain_name":"independent","created_at":"2019-02-18T02:03:26.325-08:00","display_name":"Rene Conijn","url":"https://independent.academia.edu/ReneConijn"},"attachments":[{"id":81581233,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/81581233/thumbnails/1.jpg","file_name":"23595B.pdf","download_url":"https://www.academia.edu/attachments/81581233/download_file?st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Planar_Critical_Percolation_Large_cluste.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/81581233/23595B-libre.pdf?1646233518=\u0026response-content-disposition=attachment%3B+filename%3DPlanar_Critical_Percolation_Large_cluste.pdf\u0026Expires=1732775116\u0026Signature=EOxAA3P5GtO5WLKQvDVz2pf5u0KFuihA3mO3VHk2Q73ia2Lk1SpevxMLgE55Ymy~GHClnsQ5rs6sRElv8YKmdGwY~jfgUvZHkoFR6ryz7ViJbmJ9xWi3S4UhN7Ua9OHXVGW8JOx-~R384pCDeIAK6zOgOwxArvUWNmpUfKObdWId8UALUt0-L-T-f58iftj97nnUCMFlAl9zVkr39uME~f-DdJEpSBaNrm1DCeanZq9AP3booeOavV8G72PM0mRG4PD3bkVX1zABLaruEUGuQ-icqvc82gF-vQJjEeDjOwKsDzcA-nSczx32QGwKd6bKNN5JEFXxCowlUqSJ552Rqg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":81581234,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/81581234/thumbnails/1.jpg","file_name":"23595B.pdf","download_url":"https://www.academia.edu/attachments/81581234/download_file","bulk_download_file_name":"Planar_Critical_Percolation_Large_cluste.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/81581234/23595B-libre.pdf?1646233523=\u0026response-content-disposition=attachment%3B+filename%3DPlanar_Critical_Percolation_Large_cluste.pdf\u0026Expires=1732775116\u0026Signature=WNDg4Yp6uji3nRy8p4oAeeTvxZNL7elVqxKM9wmUBlDFrrMBiIkM42nCq1fGD1M-SyxZVNz7yTcBO0KDhw~8Y1OrTLjnvKM3nAGgTzRgdWN4KQQRGoRF~bswZHCveUzHbznF1l-JsuUqKLY9~P9bDRRt1VGjxYijPbqtikY1K8HXNE1iYCFD8ZfcMy-NIF5O-oVd6sUVzCyWR~7bur0T2qYePD1DE3C6V52zhe15108Df-B4UrZlt9TUgkMkoTDtYuIlpeWjMIFgbZATSwozqitzQdRsPPfytDMEpXk6Ci6FeAqKxgd6GR5iBrMSbGNUndoBF~InXhA4EEPsFXt7gQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"}],"urls":[{"id":18152042,"url":"https://ir.cwi.nl/pub/23595/23595B.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="72794607"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/72794607/The_expected_number_of_critical_percolation_clusters_intersecting_a_line_segment"><img alt="Research paper thumbnail of The expected number of critical percolation clusters intersecting a line segment" class="work-thumbnail" src="https://attachments.academia-assets.com/81579813/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/72794607/The_expected_number_of_critical_percolation_clusters_intersecting_a_line_segment">The expected number of critical percolation clusters intersecting a line segment</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We study critical percolation on a regular planar lattice. Let E_G(n) be the expected number of o...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We study critical percolation on a regular planar lattice. Let E_G(n) be the expected number of open clusters intersecting or hitting the line segment [0,n]. (For the subscript G we either take H, when we restrict to the upper halfplane, or C, when we consider the full lattice). Cardy (2001) (see also Yu, Saleur and Haas (2008)) derived heuristically that E_H(n) = An + √(3)/4π(n) + o((n)), where A is some constant. Recently Kovács, Iglói and Cardy (2012) derived heuristically (as a special case of a more general formula) that a similar result holds for E_C(n) with the constant √(3)/4π replaced by 5√(3)/32π. In this paper we give, for site percolation on the triangular lattice, a rigorous proof for the formula of E_H(n) above, and a rigorous upper bound for the prefactor of the logarithm in the formula of E_C(n).</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="6345b816d4006bd8c0d4056572635c92" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":81579813,"asset_id":72794607,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/81579813/download_file?st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="72794607"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="72794607"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 72794607; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=72794607]").text(description); $(".js-view-count[data-work-id=72794607]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 72794607; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='72794607']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 72794607, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "6345b816d4006bd8c0d4056572635c92" } } $('.js-work-strip[data-work-id=72794607]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":72794607,"title":"The expected number of critical percolation clusters intersecting a line segment","translated_title":"","metadata":{"abstract":"We study critical percolation on a regular planar lattice. Let E_G(n) be the expected number of open clusters intersecting or hitting the line segment [0,n]. (For the subscript G we either take H, when we restrict to the upper halfplane, or C, when we consider the full lattice). Cardy (2001) (see also Yu, Saleur and Haas (2008)) derived heuristically that E_H(n) = An + √(3)/4π(n) + o((n)), where A is some constant. Recently Kovács, Iglói and Cardy (2012) derived heuristically (as a special case of a more general formula) that a similar result holds for E_C(n) with the constant √(3)/4π replaced by 5√(3)/32π. In this paper we give, for site percolation on the triangular lattice, a rigorous proof for the formula of E_H(n) above, and a rigorous upper bound for the prefactor of the logarithm in the formula of E_C(n).","publication_date":{"day":29,"month":3,"year":2016,"errors":{}}},"translated_abstract":"We study critical percolation on a regular planar lattice. Let E_G(n) be the expected number of open clusters intersecting or hitting the line segment [0,n]. (For the subscript G we either take H, when we restrict to the upper halfplane, or C, when we consider the full lattice). Cardy (2001) (see also Yu, Saleur and Haas (2008)) derived heuristically that E_H(n) = An + √(3)/4π(n) + o((n)), where A is some constant. Recently Kovács, Iglói and Cardy (2012) derived heuristically (as a special case of a more general formula) that a similar result holds for E_C(n) with the constant √(3)/4π replaced by 5√(3)/32π. In this paper we give, for site percolation on the triangular lattice, a rigorous proof for the formula of E_H(n) above, and a rigorous upper bound for the prefactor of the logarithm in the formula of E_C(n).","internal_url":"https://www.academia.edu/72794607/The_expected_number_of_critical_percolation_clusters_intersecting_a_line_segment","translated_internal_url":"","created_at":"2022-03-02T06:29:55.049-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":102711148,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":81579813,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/81579813/thumbnails/1.jpg","file_name":"1505.08046v2.pdf","download_url":"https://www.academia.edu/attachments/81579813/download_file?st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_expected_number_of_critical_percolat.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/81579813/1505.08046v2-libre.pdf?1646231818=\u0026response-content-disposition=attachment%3B+filename%3DThe_expected_number_of_critical_percolat.pdf\u0026Expires=1732775116\u0026Signature=ShuYSqhZBSLxxzqDWp1bcz7c8g803InAg74mcTg~ychxMUh8qpItPP5iqCS-iovF2JOavdmq8EJVKM6Hv-kpKR4FrsdX9Tf7F1R3DLFNIgJZsmjI1X-WNOa0Rg0GWiholE4yrimOPksZHKF22YCIUsrSfDtr4j5eZrr-iYjz7G6Bmb58CVBdlIC-qYO5DctVZvJ-C77eHuNWYF2LN6lC0Y5xZ6uoM2hgtwpArOVEP1Ewa-pVRzKJRrMqfyWVSdp7RZ8gAeACGKcK5fd0OGkwdj5bD11TJsMqmTgbuGv4~vSwrNLCQw4BmcOmDzeQfhIRiW31NpBU926kxKl0z6RpMw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"The_expected_number_of_critical_percolation_clusters_intersecting_a_line_segment","translated_slug":"","page_count":12,"language":"en","content_type":"Work","owner":{"id":102711148,"first_name":"Rene","middle_initials":null,"last_name":"Conijn","page_name":"ReneConijn","domain_name":"independent","created_at":"2019-02-18T02:03:26.325-08:00","display_name":"Rene Conijn","url":"https://independent.academia.edu/ReneConijn"},"attachments":[{"id":81579813,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/81579813/thumbnails/1.jpg","file_name":"1505.08046v2.pdf","download_url":"https://www.academia.edu/attachments/81579813/download_file?st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_expected_number_of_critical_percolat.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/81579813/1505.08046v2-libre.pdf?1646231818=\u0026response-content-disposition=attachment%3B+filename%3DThe_expected_number_of_critical_percolat.pdf\u0026Expires=1732775116\u0026Signature=ShuYSqhZBSLxxzqDWp1bcz7c8g803InAg74mcTg~ychxMUh8qpItPP5iqCS-iovF2JOavdmq8EJVKM6Hv-kpKR4FrsdX9Tf7F1R3DLFNIgJZsmjI1X-WNOa0Rg0GWiholE4yrimOPksZHKF22YCIUsrSfDtr4j5eZrr-iYjz7G6Bmb58CVBdlIC-qYO5DctVZvJ-C77eHuNWYF2LN6lC0Y5xZ6uoM2hgtwpArOVEP1Ewa-pVRzKJRrMqfyWVSdp7RZ8gAeACGKcK5fd0OGkwdj5bD11TJsMqmTgbuGv4~vSwrNLCQw4BmcOmDzeQfhIRiW31NpBU926kxKl0z6RpMw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":81579812,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/81579812/thumbnails/1.jpg","file_name":"1505.08046v2.pdf","download_url":"https://www.academia.edu/attachments/81579812/download_file","bulk_download_file_name":"The_expected_number_of_critical_percolat.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/81579812/1505.08046v2-libre.pdf?1646231817=\u0026response-content-disposition=attachment%3B+filename%3DThe_expected_number_of_critical_percolat.pdf\u0026Expires=1732775116\u0026Signature=JLTVr3~r6YWucQPuI65OK7sOXvCYt7XebE0NT0LuVM4YPHKR~WpVP2u6jAfoWsPfoLLwpXL39dNhVnyoh-U9dG2kC2jMslW1nPIE-V6RxIscruAmkw~yLUh9tzX3L3UVgxVECZDVz4wDwhYpt0iK2nxAiq7foXOOZHfpqP6ZeQlPrAOJ6RyzVj3f2za7Of2QLjdOf7lFWrqqQI6p2Ms5ZoU43TvedtkJKlMWNtf6bkQaFbETs65KZpCAovfQtG9zTZaeiR01ph3PTwwZut~xaYDUR1-OoIV~RyO3Vk54zMr7o6jK-r3ZX8fOLh5Kuz~b2kuQWW62emksGcWbklilRg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":892,"name":"Statistics","url":"https://www.academia.edu/Documents/in/Statistics"}],"urls":[{"id":18150951,"url":"https://arxiv.org/pdf/1505.08046v2.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="62924981"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/62924981/Factorization_formulas_for_2D_critical_percolation_revisited"><img alt="Research paper thumbnail of Factorization formulas for 2D critical percolation, revisited" class="work-thumbnail" src="https://attachments.academia-assets.com/75531425/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/62924981/Factorization_formulas_for_2D_critical_percolation_revisited">Factorization formulas for 2D critical percolation, revisited</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We consider critical site percolation on the triangular lattice in the upper half-plane. Let u1,u...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We consider critical site percolation on the triangular lattice in the upper half-plane. Let u1,u2 be two sites on the boundary and w a site in the interior. It was predicted by Simmons et al. (2007) that the ratio P(nu1↔nu2↔nw)2/P(nu1↔nu2)⋅P(nu1↔nw)⋅P(nu2↔nw) converges to KF as n→∞, where x↔y denotes that x and y are in the same cluster, and KF is a constant. Beliaev and Izyurov (2012) proved an analog of this in the scaling limit. We prove, using their result and a generalized coupling argument, the earlier mentioned prediction. Furthermore we prove a factorization formula for P(nu2↔[nu1,nu1+s];nw↔[nu1,nu1+s]), where s&gt;0.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="eab91ff78cfee043c3de560fc720fe8b" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":75531425,"asset_id":62924981,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/75531425/download_file?st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="62924981"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="62924981"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 62924981; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=62924981]").text(description); $(".js-view-count[data-work-id=62924981]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 62924981; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='62924981']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 62924981, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "eab91ff78cfee043c3de560fc720fe8b" } } $('.js-work-strip[data-work-id=62924981]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":62924981,"title":"Factorization formulas for 2D critical percolation, revisited","translated_title":"","metadata":{"abstract":"We consider critical site percolation on the triangular lattice in the upper half-plane. Let u1,u2 be two sites on the boundary and w a site in the interior. It was predicted by Simmons et al. (2007) that the ratio P(nu1↔nu2↔nw)2/P(nu1↔nu2)⋅P(nu1↔nw)⋅P(nu2↔nw) converges to KF as n→∞, where x↔y denotes that x and y are in the same cluster, and KF is a constant. Beliaev and Izyurov (2012) proved an analog of this in the scaling limit. We prove, using their result and a generalized coupling argument, the earlier mentioned prediction. Furthermore we prove a factorization formula for P(nu2↔[nu1,nu1+s];nw↔[nu1,nu1+s]), where s\u0026gt;0.","publication_date":{"day":null,"month":null,"year":2015,"errors":{}}},"translated_abstract":"We consider critical site percolation on the triangular lattice in the upper half-plane. Let u1,u2 be two sites on the boundary and w a site in the interior. It was predicted by Simmons et al. (2007) that the ratio P(nu1↔nu2↔nw)2/P(nu1↔nu2)⋅P(nu1↔nw)⋅P(nu2↔nw) converges to KF as n→∞, where x↔y denotes that x and y are in the same cluster, and KF is a constant. Beliaev and Izyurov (2012) proved an analog of this in the scaling limit. We prove, using their result and a generalized coupling argument, the earlier mentioned prediction. Furthermore we prove a factorization formula for P(nu2↔[nu1,nu1+s];nw↔[nu1,nu1+s]), where s\u0026gt;0.","internal_url":"https://www.academia.edu/62924981/Factorization_formulas_for_2D_critical_percolation_revisited","translated_internal_url":"","created_at":"2021-12-01T22:00:50.931-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":102711148,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":75531425,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/75531425/thumbnails/1.jpg","file_name":"1502.pdf","download_url":"https://www.academia.edu/attachments/75531425/download_file?st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Factorization_formulas_for_2D_critical_p.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/75531425/1502-libre.pdf?1638577793=\u0026response-content-disposition=attachment%3B+filename%3DFactorization_formulas_for_2D_critical_p.pdf\u0026Expires=1732775116\u0026Signature=E5vhUjEOux~hb5NWCcJ~u2HL7zgzkV6~oXHwSskNjage5m6ckTJB3BJiW6-m7XxRYaEJBByIogwuahuJ1XcLiztvNZxgBbwJLwAUqBxOl9Ux5KA3d5bdjI392i2nnXwzrd~0pkyaDLDH2uZUV4~VwpBJ9RTt4-SqoINUOAQW4jpI2FWHdjtCjE9SlthiSR8JbRCVRGBWdskKzWBnnDo2i9vxHuvFvsynoNgoBBRBVWE4iu-XZAKhW62EI6AgWIuHES2kdYxb6oAmEsFonASsksZ4BxNSphErlFSMXetbFLmMxoVYrNe7qL0J7s-Lz~iuzr4jJUTNpbkoyZNJaNCcxA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Factorization_formulas_for_2D_critical_percolation_revisited","translated_slug":"","page_count":17,"language":"en","content_type":"Work","owner":{"id":102711148,"first_name":"Rene","middle_initials":null,"last_name":"Conijn","page_name":"ReneConijn","domain_name":"independent","created_at":"2019-02-18T02:03:26.325-08:00","display_name":"Rene Conijn","url":"https://independent.academia.edu/ReneConijn"},"attachments":[{"id":75531425,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/75531425/thumbnails/1.jpg","file_name":"1502.pdf","download_url":"https://www.academia.edu/attachments/75531425/download_file?st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Factorization_formulas_for_2D_critical_p.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/75531425/1502-libre.pdf?1638577793=\u0026response-content-disposition=attachment%3B+filename%3DFactorization_formulas_for_2D_critical_p.pdf\u0026Expires=1732775116\u0026Signature=E5vhUjEOux~hb5NWCcJ~u2HL7zgzkV6~oXHwSskNjage5m6ckTJB3BJiW6-m7XxRYaEJBByIogwuahuJ1XcLiztvNZxgBbwJLwAUqBxOl9Ux5KA3d5bdjI392i2nnXwzrd~0pkyaDLDH2uZUV4~VwpBJ9RTt4-SqoINUOAQW4jpI2FWHdjtCjE9SlthiSR8JbRCVRGBWdskKzWBnnDo2i9vxHuvFvsynoNgoBBRBVWE4iu-XZAKhW62EI6AgWIuHES2kdYxb6oAmEsFonASsksZ4BxNSphErlFSMXetbFLmMxoVYrNe7qL0J7s-Lz~iuzr4jJUTNpbkoyZNJaNCcxA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":75531424,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/75531424/thumbnails/1.jpg","file_name":"1502.pdf","download_url":"https://www.academia.edu/attachments/75531424/download_file","bulk_download_file_name":"Factorization_formulas_for_2D_critical_p.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/75531424/1502-libre.pdf?1638577794=\u0026response-content-disposition=attachment%3B+filename%3DFactorization_formulas_for_2D_critical_p.pdf\u0026Expires=1732775116\u0026Signature=WuUFBQhpupNyJcPwDs4OZMiz9IktV19G4FeBBh8KOjZjVNf9z5ZujTZYt1Me36RjKRI4AYj0YWYWBq~1Vih4P~j1EGyiFwZcJ4bhePvtD07s03MKqzZBbtPheKMRT2QfxwgmYoJbhA2-zW57Af7kQshFGm51luLjoIXr3pQ6L~6fVxPRHKWcp6AmAnBTDw8uL3Mlwi~Qe2iw~F9MLcAtjWWcvlbqJRQRLlCmFg5OQJrjQQXrDwJG5fNVWYujuGYaHjhHnCx9Q5l8ZVjqIODsQYo3JQ~7kdMXymdrEkwKqKgDFBBqHHla~na-2fTz6QIRHFqNNsE1N4Grxwo3tF039w__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":892,"name":"Statistics","url":"https://www.academia.edu/Documents/in/Statistics"},{"id":3079415,"name":"Finance and Investment Banking","url":"https://www.academia.edu/Documents/in/Finance_and_Investment_Banking"}],"urls":[{"id":14681358,"url":"http://export.arxiv.org/pdf/1502.04387"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="54335823"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/54335823/Factorization_Formulas_for_2D_Critical_Percolation_Revisited"><img alt="Research paper thumbnail of Factorization Formulas for $2D$ Critical Percolation, Revisited" class="work-thumbnail" src="https://attachments.academia-assets.com/70749716/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/54335823/Factorization_Formulas_for_2D_Critical_Percolation_Revisited">Factorization Formulas for $2D$ Critical Percolation, Revisited</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We consider critical site percolation on the triangular lattice in the upper half-plane. Let $u_1...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We consider critical site percolation on the triangular lattice in the upper half-plane. Let $u_1, u_2$ be two sites on the boundary and $w$ a site in the interior of the half-plane. It was predicted by Simmons, Kleban and Ziff in a paper from 2007 that the ratio $\mathbb{P}(nu_1 \leftrightarrow nu_2 \leftrightarrow nw)^{2}\,/\,\mathbb{P}(nu_1 \leftrightarrow nu_2)\cdot\mathbb{P}(nu_1 \leftrightarrow nw)\cdot\mathbb{P}(nu_2 \leftrightarrow nw)$ converges to $K_F$ as $n \to \infty$, where $x\leftrightarrow y$ denotes the event that $x$ and $y$ are in the same open cluster, and $K_F$ is an explicitly known constant. Beliaev and Izyurov proved in a paper in 2012 an analog of this factorization in the scaling limit. We prove, using their result and a generalized coupling argument, the earlier mentioned prediction. Furthermore we prove a factorization formula for the probability $\mathbb{P}(nu_2 \leftrightarrow [nu_1,nu_1+s];\, nw \leftrightarrow [nu_1,nu_1+s])$, where $s>0$.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="81cd6188f41efe1529fba52ad1898a4b" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":70749716,"asset_id":54335823,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/70749716/download_file?st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="54335823"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="54335823"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 54335823; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=54335823]").text(description); $(".js-view-count[data-work-id=54335823]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 54335823; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='54335823']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 54335823, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "81cd6188f41efe1529fba52ad1898a4b" } } $('.js-work-strip[data-work-id=54335823]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":54335823,"title":"Factorization Formulas for $2D$ Critical Percolation, Revisited","translated_title":"","metadata":{"abstract":"We consider critical site percolation on the triangular lattice in the upper half-plane. Let $u_1, u_2$ be two sites on the boundary and $w$ a site in the interior of the half-plane. It was predicted by Simmons, Kleban and Ziff in a paper from 2007 that the ratio $\\mathbb{P}(nu_1 \\leftrightarrow nu_2 \\leftrightarrow nw)^{2}\\,/\\,\\mathbb{P}(nu_1 \\leftrightarrow nu_2)\\cdot\\mathbb{P}(nu_1 \\leftrightarrow nw)\\cdot\\mathbb{P}(nu_2 \\leftrightarrow nw)$ converges to $K_F$ as $n \\to \\infty$, where $x\\leftrightarrow y$ denotes the event that $x$ and $y$ are in the same open cluster, and $K_F$ is an explicitly known constant. Beliaev and Izyurov proved in a paper in 2012 an analog of this factorization in the scaling limit. We prove, using their result and a generalized coupling argument, the earlier mentioned prediction. Furthermore we prove a factorization formula for the probability $\\mathbb{P}(nu_2 \\leftrightarrow [nu_1,nu_1+s];\\, nw \\leftrightarrow [nu_1,nu_1+s])$, where $s\u003e0$.","publication_date":{"day":15,"month":2,"year":2015,"errors":{}}},"translated_abstract":"We consider critical site percolation on the triangular lattice in the upper half-plane. Let $u_1, u_2$ be two sites on the boundary and $w$ a site in the interior of the half-plane. It was predicted by Simmons, Kleban and Ziff in a paper from 2007 that the ratio $\\mathbb{P}(nu_1 \\leftrightarrow nu_2 \\leftrightarrow nw)^{2}\\,/\\,\\mathbb{P}(nu_1 \\leftrightarrow nu_2)\\cdot\\mathbb{P}(nu_1 \\leftrightarrow nw)\\cdot\\mathbb{P}(nu_2 \\leftrightarrow nw)$ converges to $K_F$ as $n \\to \\infty$, where $x\\leftrightarrow y$ denotes the event that $x$ and $y$ are in the same open cluster, and $K_F$ is an explicitly known constant. Beliaev and Izyurov proved in a paper in 2012 an analog of this factorization in the scaling limit. We prove, using their result and a generalized coupling argument, the earlier mentioned prediction. Furthermore we prove a factorization formula for the probability $\\mathbb{P}(nu_2 \\leftrightarrow [nu_1,nu_1+s];\\, nw \\leftrightarrow [nu_1,nu_1+s])$, where $s\u003e0$.","internal_url":"https://www.academia.edu/54335823/Factorization_Formulas_for_2D_Critical_Percolation_Revisited","translated_internal_url":"","created_at":"2021-09-30T07:52:09.502-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":102711148,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":70749716,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/70749716/thumbnails/1.jpg","file_name":"1502.04387.pdf","download_url":"https://www.academia.edu/attachments/70749716/download_file?st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Factorization_Formulas_for_2D_Critical_P.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/70749716/1502.04387-libre.pdf?1633018236=\u0026response-content-disposition=attachment%3B+filename%3DFactorization_Formulas_for_2D_Critical_P.pdf\u0026Expires=1732775116\u0026Signature=PJDjbscYVMc0hKiq801qggTtW5cj5SDWMzm9hYs9SDmzvALbhmlKiLCZnD5yZlR~FxzBK9jMM42jHbofw~A-mKgZtsFrkiLbK6Pm3oMrvO4rCyz95BqtHTRIS0HGirlI0ut6FSs6mz~4NWt2NZG6fyOyxiCJac82QprgvkIZ1hjvUynzydEdpjL0HShipSFbzKx83G2-uv0r8CuWyBKYNZETEll8pFPQ1ganwJPCmecZChSTe8msFRNSfaZbpjc7cpE1IOzc4Sr0hhK8TInaBAt-kPFZdBKg-6r~LY~f9HLHVY9TW~VoJwFnsDde5hP5F7kh2aZc6nrAfI7j3~Vq1A__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Factorization_Formulas_for_2D_Critical_Percolation_Revisited","translated_slug":"","page_count":17,"language":"en","content_type":"Work","owner":{"id":102711148,"first_name":"Rene","middle_initials":null,"last_name":"Conijn","page_name":"ReneConijn","domain_name":"independent","created_at":"2019-02-18T02:03:26.325-08:00","display_name":"Rene Conijn","url":"https://independent.academia.edu/ReneConijn"},"attachments":[{"id":70749716,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/70749716/thumbnails/1.jpg","file_name":"1502.04387.pdf","download_url":"https://www.academia.edu/attachments/70749716/download_file?st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Factorization_Formulas_for_2D_Critical_P.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/70749716/1502.04387-libre.pdf?1633018236=\u0026response-content-disposition=attachment%3B+filename%3DFactorization_Formulas_for_2D_Critical_P.pdf\u0026Expires=1732775116\u0026Signature=PJDjbscYVMc0hKiq801qggTtW5cj5SDWMzm9hYs9SDmzvALbhmlKiLCZnD5yZlR~FxzBK9jMM42jHbofw~A-mKgZtsFrkiLbK6Pm3oMrvO4rCyz95BqtHTRIS0HGirlI0ut6FSs6mz~4NWt2NZG6fyOyxiCJac82QprgvkIZ1hjvUynzydEdpjL0HShipSFbzKx83G2-uv0r8CuWyBKYNZETEll8pFPQ1ganwJPCmecZChSTe8msFRNSfaZbpjc7cpE1IOzc4Sr0hhK8TInaBAt-kPFZdBKg-6r~LY~f9HLHVY9TW~VoJwFnsDde5hP5F7kh2aZc6nrAfI7j3~Vq1A__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[],"urls":[{"id":11823191,"url":"http://arxiv.org/abs/1502.04387"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="54335806"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/54335806/The_expected_number_of_critical_percolation_clusters_intersecting_a_line_segment"><img alt="Research paper thumbnail of The expected number of critical percolation clusters intersecting a line segment" class="work-thumbnail" src="https://attachments.academia-assets.com/70749702/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/54335806/The_expected_number_of_critical_percolation_clusters_intersecting_a_line_segment">The expected number of critical percolation clusters intersecting a line segment</a></div><div class="wp-workCard_item"><span>Electronic Communications in Probability</span><span>, 2016</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2b50175132cb4b56941911e440c47f0e" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":70749702,"asset_id":54335806,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/70749702/download_file?st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="54335806"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="54335806"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 54335806; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=54335806]").text(description); $(".js-view-count[data-work-id=54335806]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 54335806; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='54335806']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 54335806, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "2b50175132cb4b56941911e440c47f0e" } } $('.js-work-strip[data-work-id=54335806]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":54335806,"title":"The expected number of critical percolation clusters intersecting a line segment","translated_title":"","metadata":{"publisher":"Institute of Mathematical Statistics","grobid_abstract":"We study critical percolation on a regular planar lattice. Let E G (n) be the expected number of open clusters intersecting or hitting the line segment [0, n]. (For the subscript G we either take H, when we restrict to the upper halfplane, or C, when we consider the full lattice). Cardy [Car01] (see also Yu, Saleur and Haas [YSH08]) derived heuristically that E H (n) = An + √ 3 4π log(n) + o(log(n)), where A is some constant. Recently Kovács, Iglói and Cardy derived in [KIC12] heuristically (as a special case of a more general formula) that a similar result holds for E C (n) with the constant √ 3 4π replaced by 5 √ 3 32π. In this paper we give, for site percolation on the triangular lattice, a rigorous proof for the formula of E H (n) above, and a rigorous upper bound for the prefactor of the logarithm in the formula of E C (n).","publication_date":{"day":null,"month":null,"year":2016,"errors":{}},"publication_name":"Electronic Communications in Probability","grobid_abstract_attachment_id":70749702},"translated_abstract":null,"internal_url":"https://www.academia.edu/54335806/The_expected_number_of_critical_percolation_clusters_intersecting_a_line_segment","translated_internal_url":"","created_at":"2021-09-30T07:52:07.499-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":102711148,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":70749702,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/70749702/thumbnails/1.jpg","file_name":"1505.pdf","download_url":"https://www.academia.edu/attachments/70749702/download_file?st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_expected_number_of_critical_percolat.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/70749702/1505-libre.pdf?1633018235=\u0026response-content-disposition=attachment%3B+filename%3DThe_expected_number_of_critical_percolat.pdf\u0026Expires=1732775116\u0026Signature=Wr6rDiGYA8AXhiFvRbMUg4IH-IAkhEi8kx1ff1e7LXs92XdA6wAoBbLEllpRGVsD3lZ79upMMiULqlIuyztT1TLem0AubSgHtvFgttrwy9LEpGdw9pk7A8oCrfMZlNaqLH2bzZHAaABKnaawwY5emWa56M4oA-J0DasAH-K9~MYSoZMgs3z5AhCCzMo3W81pFZfaTUVo3iTRV72xdVEL2eUos8JbMapLUturcu3IsieIBk6RETd6uw0alUiGr6P1PEhawGimDvgKRpqOtTXQVxgxQ~p5Rlhu8npTI7-3LCOmjYHzRsFBY~~SoIYHYERgDvCi4G5Vv~ikocduo7oXcQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"The_expected_number_of_critical_percolation_clusters_intersecting_a_line_segment","translated_slug":"","page_count":12,"language":"en","content_type":"Work","owner":{"id":102711148,"first_name":"Rene","middle_initials":null,"last_name":"Conijn","page_name":"ReneConijn","domain_name":"independent","created_at":"2019-02-18T02:03:26.325-08:00","display_name":"Rene Conijn","url":"https://independent.academia.edu/ReneConijn"},"attachments":[{"id":70749702,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/70749702/thumbnails/1.jpg","file_name":"1505.pdf","download_url":"https://www.academia.edu/attachments/70749702/download_file?st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_expected_number_of_critical_percolat.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/70749702/1505-libre.pdf?1633018235=\u0026response-content-disposition=attachment%3B+filename%3DThe_expected_number_of_critical_percolat.pdf\u0026Expires=1732775116\u0026Signature=Wr6rDiGYA8AXhiFvRbMUg4IH-IAkhEi8kx1ff1e7LXs92XdA6wAoBbLEllpRGVsD3lZ79upMMiULqlIuyztT1TLem0AubSgHtvFgttrwy9LEpGdw9pk7A8oCrfMZlNaqLH2bzZHAaABKnaawwY5emWa56M4oA-J0DasAH-K9~MYSoZMgs3z5AhCCzMo3W81pFZfaTUVo3iTRV72xdVEL2eUos8JbMapLUturcu3IsieIBk6RETd6uw0alUiGr6P1PEhawGimDvgKRpqOtTXQVxgxQ~p5Rlhu8npTI7-3LCOmjYHzRsFBY~~SoIYHYERgDvCi4G5Vv~ikocduo7oXcQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":892,"name":"Statistics","url":"https://www.academia.edu/Documents/in/Statistics"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="54335787"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/54335787/Conformal_Measure_Ensembles_for_Percolation_and_the_FK_Ising_model"><img alt="Research paper thumbnail of Conformal Measure Ensembles for Percolation and the FK-Ising model" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/54335787/Conformal_Measure_Ensembles_for_Percolation_and_the_FK_Ising_model">Conformal Measure Ensembles for Percolation and the FK-Ising model</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Under some general assumptions we construct the scaling limit of open clusters and their associat...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Under some general assumptions we construct the scaling limit of open clusters and their associated counting measures in a class of two dimensional percolation models. Our results apply, in particular, to critical Bernoulli site percolation on the triangular lattice. We also provide conditional results for the critical FK-Ising model on the square lattice. Fundamental properties of the scaling limit, such as conformal covariance, are explored. Applications such as the scaling limit of the largest cluster in a bounded domain and a geometric representation of the magnetization field for the critical Ising model are presented.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="54335787"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="54335787"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 54335787; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=54335787]").text(description); $(".js-view-count[data-work-id=54335787]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 54335787; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='54335787']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 54335787, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=54335787]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":54335787,"title":"Conformal Measure Ensembles for Percolation and the FK-Ising model","translated_title":"","metadata":{"abstract":"Under some general assumptions we construct the scaling limit of open clusters and their associated counting measures in a class of two dimensional percolation models. Our results apply, in particular, to critical Bernoulli site percolation on the triangular lattice. We also provide conditional results for the critical FK-Ising model on the square lattice. Fundamental properties of the scaling limit, such as conformal covariance, are explored. Applications such as the scaling limit of the largest cluster in a bounded domain and a geometric representation of the magnetization field for the critical Ising model are presented."},"translated_abstract":"Under some general assumptions we construct the scaling limit of open clusters and their associated counting measures in a class of two dimensional percolation models. Our results apply, in particular, to critical Bernoulli site percolation on the triangular lattice. We also provide conditional results for the critical FK-Ising model on the square lattice. Fundamental properties of the scaling limit, such as conformal covariance, are explored. Applications such as the scaling limit of the largest cluster in a bounded domain and a geometric representation of the magnetization field for the critical Ising model are presented.","internal_url":"https://www.academia.edu/54335787/Conformal_Measure_Ensembles_for_Percolation_and_the_FK_Ising_model","translated_internal_url":"","created_at":"2021-09-30T07:52:05.657-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":102711148,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[],"slug":"Conformal_Measure_Ensembles_for_Percolation_and_the_FK_Ising_model","translated_slug":"","page_count":null,"language":"en","content_type":"Work","owner":{"id":102711148,"first_name":"Rene","middle_initials":null,"last_name":"Conijn","page_name":"ReneConijn","domain_name":"independent","created_at":"2019-02-18T02:03:26.325-08:00","display_name":"Rene Conijn","url":"https://independent.academia.edu/ReneConijn"},"attachments":[],"research_interests":[],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="54335771"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/54335771/On_the_size_of_the_largest_cluster_in_2D_critical_percolation"><img alt="Research paper thumbnail of On the size of the largest cluster in 2D critical percolation" class="work-thumbnail" src="https://attachments.academia-assets.com/70749697/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/54335771/On_the_size_of_the_largest_cluster_in_2D_critical_percolation">On the size of the largest cluster in 2D critical percolation</a></div><div class="wp-workCard_item"><span>Electronic Communications in Probability</span><span>, 2012</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="d47c86c6c6ab6a5adc5eef65f7b08a2a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":70749697,"asset_id":54335771,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/70749697/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="54335771"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="54335771"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 54335771; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=54335771]").text(description); $(".js-view-count[data-work-id=54335771]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 54335771; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='54335771']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 54335771, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "d47c86c6c6ab6a5adc5eef65f7b08a2a" } } $('.js-work-strip[data-work-id=54335771]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":54335771,"title":"On the size of the largest cluster in 2D critical percolation","translated_title":"","metadata":{"publisher":"Institute of Mathematical Statistics","grobid_abstract":"We consider (near-)critical percolation on the square lattice. Let Mn be the size of the largest open cluster contained in the box [−n, n] 2 , and let π(n) be the probability that there is an open path from O to the boundary of the box. It is well-known (see [BCKS01]) that for all 0 \u003c a \u003c b the probability that Mn is smaller than an 2 π(n) and the probability that Mn is larger than bn 2 π(n) are bounded away from 0 as n → ∞. It is a natural question, which arises for instance in the study of so-called frozenpercolation processes, if a similar result holds for the probability that Mn is between an 2 π(n) and bn 2 π(n). By a suitable partition of the box, and a careful construction involving the building blocks, we show that the answer to this question is affirmative. The 'sublinearity' of 1/π(n) appears to be essential for the argument.","publication_date":{"day":null,"month":null,"year":2012,"errors":{}},"publication_name":"Electronic Communications in Probability","grobid_abstract_attachment_id":70749697},"translated_abstract":null,"internal_url":"https://www.academia.edu/54335771/On_the_size_of_the_largest_cluster_in_2D_critical_percolation","translated_internal_url":"","created_at":"2021-09-30T07:52:02.781-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":102711148,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":70749697,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/70749697/thumbnails/1.jpg","file_name":"1208.pdf","download_url":"https://www.academia.edu/attachments/70749697/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_the_size_of_the_largest_cluster_in_2D.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/70749697/1208-libre.pdf?1633018237=\u0026response-content-disposition=attachment%3B+filename%3DOn_the_size_of_the_largest_cluster_in_2D.pdf\u0026Expires=1732775116\u0026Signature=cEMirdpdJsfO3opjeEu6pZDXFX0xLuCPxIpKciWPiuWot4dZE8cBECgr32nB63G8GCN8uG1mo9cbcoTrQb28vAdignbD5m0E~NTsygH6l3polLuZvIqyZBdvdhcIlU5wBIqAx~HGvc8jY0ihJhshKs~bRE3b5tmKp2Ra5bGvHW9nKun34NkYMUwJeLvVXvhcaY-wIqfae9jUG8S7H2BkugeBeTKK4HE9avrqN7MaU0uE12i2g4uAMyEYF5wXbMjqFgoyOxcey77FdGJvugeBr5cd-sPZZ55EjYuwwi01sSevuii~2~a4AVkxGuOWGQNd4Rc146DxgzN8n96w3rSigw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"On_the_size_of_the_largest_cluster_in_2D_critical_percolation","translated_slug":"","page_count":12,"language":"en","content_type":"Work","owner":{"id":102711148,"first_name":"Rene","middle_initials":null,"last_name":"Conijn","page_name":"ReneConijn","domain_name":"independent","created_at":"2019-02-18T02:03:26.325-08:00","display_name":"Rene Conijn","url":"https://independent.academia.edu/ReneConijn"},"attachments":[{"id":70749697,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/70749697/thumbnails/1.jpg","file_name":"1208.pdf","download_url":"https://www.academia.edu/attachments/70749697/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_the_size_of_the_largest_cluster_in_2D.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/70749697/1208-libre.pdf?1633018237=\u0026response-content-disposition=attachment%3B+filename%3DOn_the_size_of_the_largest_cluster_in_2D.pdf\u0026Expires=1732775117\u0026Signature=aAgW1Dvjwb8RV2hvn6gk8b3i9sD2Zl2l7aFFB8HjIXVEaC0FHPaoKGl~RKDVnDQXQ23lVs3vA4YlL9fxJxfp5ivK06cgy-gVCXsWpXHCfoMcOPS2xjuR1zHE6LdcseyTpEkq2xoDc3W1yexyqCDOwRHaVBbCJ9f5BdS4nyyIJYXkP7XJWVOc96qWktbYqGDA2pNCNIi89om2ynEv1k5qaVBC9~sqeDOdP6~sbhbkdIfFHA3QhIxaarqfSj8MG3tT1v4R-XOuTwsy~ZOPWDfN5a39KW3AO8GEm1Fu7gA6HHDvweXYHbENnc1xjq0t0T3t137jtXgO4nBX8IU6WFTXow__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":892,"name":"Statistics","url":"https://www.academia.edu/Documents/in/Statistics"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="54335511"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/54335511/The_gaps_between_the_sizes_of_large_clusters_in_2D_critical_percolation"><img alt="Research paper thumbnail of The gaps between the sizes of large clusters in 2D critical percolation" class="work-thumbnail" src="https://attachments.academia-assets.com/70749541/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/54335511/The_gaps_between_the_sizes_of_large_clusters_in_2D_critical_percolation">The gaps between the sizes of large clusters in 2D critical percolation</a></div><div class="wp-workCard_item"><span>Electronic Communications in Probability</span><span>, 2013</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="207226ba4b39c269285d728ad9f61a5a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":70749541,"asset_id":54335511,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/70749541/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="54335511"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="54335511"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 54335511; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=54335511]").text(description); $(".js-view-count[data-work-id=54335511]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 54335511; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='54335511']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 54335511, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "207226ba4b39c269285d728ad9f61a5a" } } $('.js-work-strip[data-work-id=54335511]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":54335511,"title":"The gaps between the sizes of large clusters in 2D critical percolation","translated_title":"","metadata":{"publisher":"Institute of Mathematical Statistics","grobid_abstract":"Consider critical bond percolation on a large 2n × 2n box on the square lattice. It is well-known that the size (i.e. number of vertices) of the largest open cluster is, with high probability, of order n 2 π(n), where π(n) denotes the probability that there is an open path from the center to the boundary of the box. The same result holds for the second-largest cluster, the third largest cluster etcetera. Járai showed that the differences between the sizes of these clusters is, with high probability, at least of order n 2 π(n). Although this bound was enough for his applications (to incipient infinite clusters), he believed, but had no proof, that the differences are in fact of the same order as the cluster sizes themselves, i.e. n 2 π(n). Our main result is a proof that this is indeed the case.","publication_date":{"day":null,"month":null,"year":2013,"errors":{}},"publication_name":"Electronic Communications in Probability","grobid_abstract_attachment_id":70749541},"translated_abstract":null,"internal_url":"https://www.academia.edu/54335511/The_gaps_between_the_sizes_of_large_clusters_in_2D_critical_percolation","translated_internal_url":"","created_at":"2021-09-30T07:51:20.013-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":102711148,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":70749541,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/70749541/thumbnails/1.jpg","file_name":"21868D.pdf","download_url":"https://www.academia.edu/attachments/70749541/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_gaps_between_the_sizes_of_large_clus.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/70749541/21868D-libre.pdf?1633018243=\u0026response-content-disposition=attachment%3B+filename%3DThe_gaps_between_the_sizes_of_large_clus.pdf\u0026Expires=1732775117\u0026Signature=WdJxNLVCQs-hC-y0t3~K5TMoGQQ5TUg2w99IIg-XvcRJovW4uN~RP1eCBXRa4Flx8bx6XJ0KwCdXg0ZZFd53JyJaS7AT5J7Kp1r6WWtSDgB5RITXcJ92AhP2EOIIkP4z7CcN3LfWbMQ2drwnmTOw7dnaIH2QVg0QoUC2FnsnGjvFvOHyglNLyUJFpid97oejS78wvAQhmD1Zq0uqkfL-152SV2FKsgfdKDIHacVAJAYplRWxlewHYJW0OEsuC3S96W-KbyO~fy1eFIss7GriRXqjvx~akQxJBTKmuImYwp4IUk9QLknrQaeFWlkgbUNjKKGjL-AbQSyIEgRl7hQ3SA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"The_gaps_between_the_sizes_of_large_clusters_in_2D_critical_percolation","translated_slug":"","page_count":10,"language":"en","content_type":"Work","owner":{"id":102711148,"first_name":"Rene","middle_initials":null,"last_name":"Conijn","page_name":"ReneConijn","domain_name":"independent","created_at":"2019-02-18T02:03:26.325-08:00","display_name":"Rene Conijn","url":"https://independent.academia.edu/ReneConijn"},"attachments":[{"id":70749541,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/70749541/thumbnails/1.jpg","file_name":"21868D.pdf","download_url":"https://www.academia.edu/attachments/70749541/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_gaps_between_the_sizes_of_large_clus.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/70749541/21868D-libre.pdf?1633018243=\u0026response-content-disposition=attachment%3B+filename%3DThe_gaps_between_the_sizes_of_large_clus.pdf\u0026Expires=1732775117\u0026Signature=WdJxNLVCQs-hC-y0t3~K5TMoGQQ5TUg2w99IIg-XvcRJovW4uN~RP1eCBXRa4Flx8bx6XJ0KwCdXg0ZZFd53JyJaS7AT5J7Kp1r6WWtSDgB5RITXcJ92AhP2EOIIkP4z7CcN3LfWbMQ2drwnmTOw7dnaIH2QVg0QoUC2FnsnGjvFvOHyglNLyUJFpid97oejS78wvAQhmD1Zq0uqkfL-152SV2FKsgfdKDIHacVAJAYplRWxlewHYJW0OEsuC3S96W-KbyO~fy1eFIss7GriRXqjvx~akQxJBTKmuImYwp4IUk9QLknrQaeFWlkgbUNjKKGjL-AbQSyIEgRl7hQ3SA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":892,"name":"Statistics","url":"https://www.academia.edu/Documents/in/Statistics"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> </div><div class="profile--tab_content_container js-tab-pane tab-pane" data-section-id="12079407" id="papers"><div class="js-work-strip profile--work_container" data-work-id="84336774"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/84336774/On_the_size_of_the_largest_cluster_in_2D_critical_percolation"><img alt="Research paper thumbnail of On the size of the largest cluster in 2D critical percolation" class="work-thumbnail" src="https://attachments.academia-assets.com/89395086/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/84336774/On_the_size_of_the_largest_cluster_in_2D_critical_percolation">On the size of the largest cluster in 2D critical percolation</a></div><div class="wp-workCard_item"><span>arXiv: Probability</span><span>, Aug 20, 2012</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="c103fee90bcdf8985596c823af0663c5" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":89395086,"asset_id":84336774,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/89395086/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="84336774"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="84336774"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 84336774; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=84336774]").text(description); $(".js-view-count[data-work-id=84336774]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 84336774; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='84336774']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 84336774, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "c103fee90bcdf8985596c823af0663c5" } } $('.js-work-strip[data-work-id=84336774]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":84336774,"title":"On the size of the largest cluster in 2D critical percolation","translated_title":"","metadata":{"grobid_abstract":"We consider (near-)critical percolation on the square lattice. Let Mn be the size of the largest open cluster contained in the box [−n, n] 2 , and let π(n) be the probability that there is an open path from O to the boundary of the box. It is well-known (see [BCKS01]) that for all 0 \u003c a \u003c b the probability that Mn is smaller than an 2 π(n) and the probability that Mn is larger than bn 2 π(n) are bounded away from 0 as n → ∞. It is a natural question, which arises for instance in the study of so-called frozenpercolation processes, if a similar result holds for the probability that Mn is between an 2 π(n) and bn 2 π(n). By a suitable partition of the box, and a careful construction involving the building blocks, we show that the answer to this question is affirmative. The 'sublinearity' of 1/π(n) appears to be essential for the argument.","publication_date":{"day":20,"month":8,"year":2012,"errors":{}},"publication_name":"arXiv: Probability","grobid_abstract_attachment_id":89395086},"translated_abstract":null,"internal_url":"https://www.academia.edu/84336774/On_the_size_of_the_largest_cluster_in_2D_critical_percolation","translated_internal_url":"","created_at":"2022-08-08T09:15:41.838-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":102711148,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":89395086,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/89395086/thumbnails/1.jpg","file_name":"1208.4014.pdf","download_url":"https://www.academia.edu/attachments/89395086/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_the_size_of_the_largest_cluster_in_2D.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/89395086/1208.4014-libre.pdf?1659976532=\u0026response-content-disposition=attachment%3B+filename%3DOn_the_size_of_the_largest_cluster_in_2D.pdf\u0026Expires=1732775116\u0026Signature=YBrp6VV5dQET1cLUObUJQfKpLrtYsES26mrJXgz~oRIe~RRFb9bwjDL6-wbaIQA-SsTjkfXbBWSIcqjeRbOTW5-J3PHA0So4lHrAw5gIaTfM5sOsdDXbWAeJABLQYauA2iJZ9wknj5Yotak1-c-HK4QjNjTRGE337OYLC25gNescEHSKLj8z~mnbmDn-iEXYsTsYfKE5gS9jFzB~S1gA1JtfvN14EmQ8jUuSe3GpEZDo9T1LUgFzv2Y16uRYTWjxm3CLwPrirrM41C0Dj6zLpLHioVEifmqWbt8fj8ByGbnRGPdGTDxW0CEX5IFaJgYWqzLjRy0GggT0kTYeB311dw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"On_the_size_of_the_largest_cluster_in_2D_critical_percolation","translated_slug":"","page_count":12,"language":"en","content_type":"Work","owner":{"id":102711148,"first_name":"Rene","middle_initials":null,"last_name":"Conijn","page_name":"ReneConijn","domain_name":"independent","created_at":"2019-02-18T02:03:26.325-08:00","display_name":"Rene Conijn","url":"https://independent.academia.edu/ReneConijn"},"attachments":[{"id":89395086,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/89395086/thumbnails/1.jpg","file_name":"1208.4014.pdf","download_url":"https://www.academia.edu/attachments/89395086/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_the_size_of_the_largest_cluster_in_2D.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/89395086/1208.4014-libre.pdf?1659976532=\u0026response-content-disposition=attachment%3B+filename%3DOn_the_size_of_the_largest_cluster_in_2D.pdf\u0026Expires=1732775116\u0026Signature=YBrp6VV5dQET1cLUObUJQfKpLrtYsES26mrJXgz~oRIe~RRFb9bwjDL6-wbaIQA-SsTjkfXbBWSIcqjeRbOTW5-J3PHA0So4lHrAw5gIaTfM5sOsdDXbWAeJABLQYauA2iJZ9wknj5Yotak1-c-HK4QjNjTRGE337OYLC25gNescEHSKLj8z~mnbmDn-iEXYsTsYfKE5gS9jFzB~S1gA1JtfvN14EmQ8jUuSe3GpEZDo9T1LUgFzv2Y16uRYTWjxm3CLwPrirrM41C0Dj6zLpLHioVEifmqWbt8fj8ByGbnRGPdGTDxW0CEX5IFaJgYWqzLjRy0GggT0kTYeB311dw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":892,"name":"Statistics","url":"https://www.academia.edu/Documents/in/Statistics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":385916,"name":"Percolation threshold","url":"https://www.academia.edu/Documents/in/Percolation_threshold"}],"urls":[{"id":22726228,"url":"https://arxiv.org/pdf/1208.4014.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="72797302"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/72797302/The_gaps_between_the_sizes_of_large_clusters_in_2D_critical_percolation"><img alt="Research paper thumbnail of The gaps between the sizes of large clusters in 2D critical percolation" class="work-thumbnail" src="https://attachments.academia-assets.com/81581237/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/72797302/The_gaps_between_the_sizes_of_large_clusters_in_2D_critical_percolation">The gaps between the sizes of large clusters in 2D critical percolation</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Consider critical bond percolation on a large 2n by 2n box on the square lattice. It is well-know...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Consider critical bond percolation on a large 2n by 2n box on the square lattice. It is well-known that the size (i.e. number of vertices) of the largest open cluster is, with high probability, of order n^2 π(n), where π(n) denotes the probability that there is an open path from the center to the boundary of the box. The same result holds for the second-largest cluster, the third largest cluster etcetera. Jarai showed that the differences between the sizes of these clusters is, with high probability, at least of order √(n^2 π(n)). Although this bound was enough for his applications (to incipient infinite clusters), he believed, but had no proof, that the differences are in fact of the same order as the cluster sizes themselves, i.e. n^2 π(n). Our main result is a proof that this is indeed the case.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="633516dcc8b5ae3043acbfb4da14573f" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":81581237,"asset_id":72797302,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/81581237/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="72797302"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="72797302"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 72797302; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=72797302]").text(description); $(".js-view-count[data-work-id=72797302]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 72797302; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='72797302']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 72797302, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "633516dcc8b5ae3043acbfb4da14573f" } } $('.js-work-strip[data-work-id=72797302]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":72797302,"title":"The gaps between the sizes of large clusters in 2D critical percolation","translated_title":"","metadata":{"abstract":"Consider critical bond percolation on a large 2n by 2n box on the square lattice. It is well-known that the size (i.e. number of vertices) of the largest open cluster is, with high probability, of order n^2 π(n), where π(n) denotes the probability that there is an open path from the center to the boundary of the box. The same result holds for the second-largest cluster, the third largest cluster etcetera. Jarai showed that the differences between the sizes of these clusters is, with high probability, at least of order √(n^2 π(n)). Although this bound was enough for his applications (to incipient infinite clusters), he believed, but had no proof, that the differences are in fact of the same order as the cluster sizes themselves, i.e. n^2 π(n). Our main result is a proof that this is indeed the case.","publication_date":{"day":8,"month":10,"year":2013,"errors":{}}},"translated_abstract":"Consider critical bond percolation on a large 2n by 2n box on the square lattice. It is well-known that the size (i.e. number of vertices) of the largest open cluster is, with high probability, of order n^2 π(n), where π(n) denotes the probability that there is an open path from the center to the boundary of the box. The same result holds for the second-largest cluster, the third largest cluster etcetera. Jarai showed that the differences between the sizes of these clusters is, with high probability, at least of order √(n^2 π(n)). Although this bound was enough for his applications (to incipient infinite clusters), he believed, but had no proof, that the differences are in fact of the same order as the cluster sizes themselves, i.e. n^2 π(n). Our main result is a proof that this is indeed the case.","internal_url":"https://www.academia.edu/72797302/The_gaps_between_the_sizes_of_large_clusters_in_2D_critical_percolation","translated_internal_url":"","created_at":"2022-03-02T06:52:58.727-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":102711148,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":81581237,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/81581237/thumbnails/1.jpg","file_name":"1310.2019v1.pdf","download_url":"https://www.academia.edu/attachments/81581237/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_gaps_between_the_sizes_of_large_clus.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/81581237/1310.2019v1-libre.pdf?1646233506=\u0026response-content-disposition=attachment%3B+filename%3DThe_gaps_between_the_sizes_of_large_clus.pdf\u0026Expires=1732775116\u0026Signature=TAsVQBKrhNzHiJAI0FYjXEfnB1cP8hByQJD59oyeOfA5gpvJCHbCO3smAS6Q6p9k9FVokQlF1pnRueO4MWnwQD4gHf6tCc3nxNHH5u7E-rmY7EV3P6MRI~5kAc~OzB3KiNoIyrxK0ZZd5pxvmtE-0wPYq84CDYMzTbRRxfKJcQTSTk~e0808v2-VnC3Emn1B-vKXVCl9Zs8vO5dVEzu3RRyEMSUqB-31tJTAuuwhSSxD8JxPdWcTdXzJc9wYPDLU8tkA1gvrlJ~jxkxxR~xkKqWaY8sQz-ecmNmWbi-0U1WfYYoXEvrLzX1QadmtT9efEWRQUPMMnkwk9SfIlHOcOQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"The_gaps_between_the_sizes_of_large_clusters_in_2D_critical_percolation","translated_slug":"","page_count":10,"language":"en","content_type":"Work","owner":{"id":102711148,"first_name":"Rene","middle_initials":null,"last_name":"Conijn","page_name":"ReneConijn","domain_name":"independent","created_at":"2019-02-18T02:03:26.325-08:00","display_name":"Rene Conijn","url":"https://independent.academia.edu/ReneConijn"},"attachments":[{"id":81581237,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/81581237/thumbnails/1.jpg","file_name":"1310.2019v1.pdf","download_url":"https://www.academia.edu/attachments/81581237/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_gaps_between_the_sizes_of_large_clus.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/81581237/1310.2019v1-libre.pdf?1646233506=\u0026response-content-disposition=attachment%3B+filename%3DThe_gaps_between_the_sizes_of_large_clus.pdf\u0026Expires=1732775116\u0026Signature=TAsVQBKrhNzHiJAI0FYjXEfnB1cP8hByQJD59oyeOfA5gpvJCHbCO3smAS6Q6p9k9FVokQlF1pnRueO4MWnwQD4gHf6tCc3nxNHH5u7E-rmY7EV3P6MRI~5kAc~OzB3KiNoIyrxK0ZZd5pxvmtE-0wPYq84CDYMzTbRRxfKJcQTSTk~e0808v2-VnC3Emn1B-vKXVCl9Zs8vO5dVEzu3RRyEMSUqB-31tJTAuuwhSSxD8JxPdWcTdXzJc9wYPDLU8tkA1gvrlJ~jxkxxR~xkKqWaY8sQz-ecmNmWbi-0U1WfYYoXEvrLzX1QadmtT9efEWRQUPMMnkwk9SfIlHOcOQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":81581235,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/81581235/thumbnails/1.jpg","file_name":"1310.2019v1.pdf","download_url":"https://www.academia.edu/attachments/81581235/download_file","bulk_download_file_name":"The_gaps_between_the_sizes_of_large_clus.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/81581235/1310.2019v1-libre.pdf?1646233507=\u0026response-content-disposition=attachment%3B+filename%3DThe_gaps_between_the_sizes_of_large_clus.pdf\u0026Expires=1732775116\u0026Signature=GaZe-KJAi1cgi-N~4RPgJKr2IKkiYDsuKgSTMmmzo1BQtLv4SjnYpIcE2OmMuFKhhrXTsOuJgqOM0dbcLSKn~~jZ13-DXCE~HEj7vbug5-Y4~ww7UklfwDtozlBFuM8ACAZJuXU2vrJEnXYw-4Y3uTgOwTBzZGpF4hAgMcPgX2cgmG2skAyMxlyZ8Ip7hl4fEVXXnqXXScpZtP20H3D8KEZCmSRK6QTYCa8AtzmfvHBOCVX9B-Sm8eRCbfspYJHjmx34We9Yw0uoghPNBekjdGuJFZUrAuBe~NvM-dXHXe-TvyMezIkU574Il6X7vjIJ7jt06feLas0Nb5sywn5OAg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":892,"name":"Statistics","url":"https://www.academia.edu/Documents/in/Statistics"}],"urls":[{"id":18152043,"url":"https://arxiv.org/pdf/1310.2019v1.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="72797299"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/72797299/Planar_Critical_Percolation_Large_clusters_and_Scaling_limits"><img alt="Research paper thumbnail of Planar Critical Percolation: Large clusters and Scaling limits" class="work-thumbnail" src="https://attachments.academia-assets.com/81581233/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/72797299/Planar_Critical_Percolation_Large_clusters_and_Scaling_limits">Planar Critical Percolation: Large clusters and Scaling limits</a></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="69691eecfe3c4266d0c53ce23d04e541" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":81581233,"asset_id":72797299,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/81581233/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="72797299"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="72797299"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 72797299; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=72797299]").text(description); $(".js-view-count[data-work-id=72797299]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 72797299; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='72797299']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 72797299, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "69691eecfe3c4266d0c53ce23d04e541" } } $('.js-work-strip[data-work-id=72797299]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":72797299,"title":"Planar Critical Percolation: Large clusters and Scaling limits","translated_title":"","metadata":{"grobid_abstract":"This chapter is based on [10] with Rob van den Berg. We consider (near-)critical percolation on the square lattice. Let M n be the size of the largest open cluster contained in the box [−n, n] 2 , and let π(n) be the probability that there is an open path from O to the boundary of the box. It is well-known (see [17]) that for all 0 \u003c a \u003c b the probability that M n is smaller than an 2 π(n) and the probability that M n is larger than bn 2 π(n) are bounded away from 0 as n → ∞. It is a natural question, which arises for instance in the study of so-called frozenpercolation processes, if a similar result holds for the probability that M n is between an 2 π(n) and bn 2 π(n). By a suitable partition of the box, and a careful construction involving the building blocks, we show that the answer to this question is armative. The`sublinearity' of 1/π(n) appears to be essential for the argument. percolation and FK-Ising This chapter is based on [20] with Federico Camia and Demeter Kiss. Under some general assumptions we construct the scaling limit of open clusters and their associated counting measures in a class of two-dimensional percolation models. Our results apply, in particular, to critical Bernoulli site percolation on the triangular lattice. We also provide conditional results for the critical FK-Ising model on the square lattice. Fundamental properties of the scaling limit, such as conformal covariance, are explored. Applications such as the scaling limit of the largest cluster in a bounded domain and a geometric representation of the magnetization eld for the critical Ising model are presented.","publication_date":{"day":null,"month":null,"year":2015,"errors":{}},"grobid_abstract_attachment_id":81581233},"translated_abstract":null,"internal_url":"https://www.academia.edu/72797299/Planar_Critical_Percolation_Large_clusters_and_Scaling_limits","translated_internal_url":"","created_at":"2022-03-02T06:52:57.953-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":102711148,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":81581233,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/81581233/thumbnails/1.jpg","file_name":"23595B.pdf","download_url":"https://www.academia.edu/attachments/81581233/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Planar_Critical_Percolation_Large_cluste.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/81581233/23595B-libre.pdf?1646233518=\u0026response-content-disposition=attachment%3B+filename%3DPlanar_Critical_Percolation_Large_cluste.pdf\u0026Expires=1732775116\u0026Signature=EOxAA3P5GtO5WLKQvDVz2pf5u0KFuihA3mO3VHk2Q73ia2Lk1SpevxMLgE55Ymy~GHClnsQ5rs6sRElv8YKmdGwY~jfgUvZHkoFR6ryz7ViJbmJ9xWi3S4UhN7Ua9OHXVGW8JOx-~R384pCDeIAK6zOgOwxArvUWNmpUfKObdWId8UALUt0-L-T-f58iftj97nnUCMFlAl9zVkr39uME~f-DdJEpSBaNrm1DCeanZq9AP3booeOavV8G72PM0mRG4PD3bkVX1zABLaruEUGuQ-icqvc82gF-vQJjEeDjOwKsDzcA-nSczx32QGwKd6bKNN5JEFXxCowlUqSJ552Rqg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Planar_Critical_Percolation_Large_clusters_and_Scaling_limits","translated_slug":"","page_count":118,"language":"en","content_type":"Work","owner":{"id":102711148,"first_name":"Rene","middle_initials":null,"last_name":"Conijn","page_name":"ReneConijn","domain_name":"independent","created_at":"2019-02-18T02:03:26.325-08:00","display_name":"Rene Conijn","url":"https://independent.academia.edu/ReneConijn"},"attachments":[{"id":81581233,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/81581233/thumbnails/1.jpg","file_name":"23595B.pdf","download_url":"https://www.academia.edu/attachments/81581233/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Planar_Critical_Percolation_Large_cluste.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/81581233/23595B-libre.pdf?1646233518=\u0026response-content-disposition=attachment%3B+filename%3DPlanar_Critical_Percolation_Large_cluste.pdf\u0026Expires=1732775116\u0026Signature=EOxAA3P5GtO5WLKQvDVz2pf5u0KFuihA3mO3VHk2Q73ia2Lk1SpevxMLgE55Ymy~GHClnsQ5rs6sRElv8YKmdGwY~jfgUvZHkoFR6ryz7ViJbmJ9xWi3S4UhN7Ua9OHXVGW8JOx-~R384pCDeIAK6zOgOwxArvUWNmpUfKObdWId8UALUt0-L-T-f58iftj97nnUCMFlAl9zVkr39uME~f-DdJEpSBaNrm1DCeanZq9AP3booeOavV8G72PM0mRG4PD3bkVX1zABLaruEUGuQ-icqvc82gF-vQJjEeDjOwKsDzcA-nSczx32QGwKd6bKNN5JEFXxCowlUqSJ552Rqg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":81581234,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/81581234/thumbnails/1.jpg","file_name":"23595B.pdf","download_url":"https://www.academia.edu/attachments/81581234/download_file","bulk_download_file_name":"Planar_Critical_Percolation_Large_cluste.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/81581234/23595B-libre.pdf?1646233523=\u0026response-content-disposition=attachment%3B+filename%3DPlanar_Critical_Percolation_Large_cluste.pdf\u0026Expires=1732775116\u0026Signature=WNDg4Yp6uji3nRy8p4oAeeTvxZNL7elVqxKM9wmUBlDFrrMBiIkM42nCq1fGD1M-SyxZVNz7yTcBO0KDhw~8Y1OrTLjnvKM3nAGgTzRgdWN4KQQRGoRF~bswZHCveUzHbznF1l-JsuUqKLY9~P9bDRRt1VGjxYijPbqtikY1K8HXNE1iYCFD8ZfcMy-NIF5O-oVd6sUVzCyWR~7bur0T2qYePD1DE3C6V52zhe15108Df-B4UrZlt9TUgkMkoTDtYuIlpeWjMIFgbZATSwozqitzQdRsPPfytDMEpXk6Ci6FeAqKxgd6GR5iBrMSbGNUndoBF~InXhA4EEPsFXt7gQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"}],"urls":[{"id":18152042,"url":"https://ir.cwi.nl/pub/23595/23595B.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="72794607"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/72794607/The_expected_number_of_critical_percolation_clusters_intersecting_a_line_segment"><img alt="Research paper thumbnail of The expected number of critical percolation clusters intersecting a line segment" class="work-thumbnail" src="https://attachments.academia-assets.com/81579813/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/72794607/The_expected_number_of_critical_percolation_clusters_intersecting_a_line_segment">The expected number of critical percolation clusters intersecting a line segment</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We study critical percolation on a regular planar lattice. Let E_G(n) be the expected number of o...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We study critical percolation on a regular planar lattice. Let E_G(n) be the expected number of open clusters intersecting or hitting the line segment [0,n]. (For the subscript G we either take H, when we restrict to the upper halfplane, or C, when we consider the full lattice). Cardy (2001) (see also Yu, Saleur and Haas (2008)) derived heuristically that E_H(n) = An + √(3)/4π(n) + o((n)), where A is some constant. Recently Kovács, Iglói and Cardy (2012) derived heuristically (as a special case of a more general formula) that a similar result holds for E_C(n) with the constant √(3)/4π replaced by 5√(3)/32π. In this paper we give, for site percolation on the triangular lattice, a rigorous proof for the formula of E_H(n) above, and a rigorous upper bound for the prefactor of the logarithm in the formula of E_C(n).</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="6345b816d4006bd8c0d4056572635c92" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":81579813,"asset_id":72794607,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/81579813/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="72794607"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="72794607"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 72794607; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=72794607]").text(description); $(".js-view-count[data-work-id=72794607]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 72794607; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='72794607']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 72794607, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "6345b816d4006bd8c0d4056572635c92" } } $('.js-work-strip[data-work-id=72794607]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":72794607,"title":"The expected number of critical percolation clusters intersecting a line segment","translated_title":"","metadata":{"abstract":"We study critical percolation on a regular planar lattice. Let E_G(n) be the expected number of open clusters intersecting or hitting the line segment [0,n]. (For the subscript G we either take H, when we restrict to the upper halfplane, or C, when we consider the full lattice). Cardy (2001) (see also Yu, Saleur and Haas (2008)) derived heuristically that E_H(n) = An + √(3)/4π(n) + o((n)), where A is some constant. Recently Kovács, Iglói and Cardy (2012) derived heuristically (as a special case of a more general formula) that a similar result holds for E_C(n) with the constant √(3)/4π replaced by 5√(3)/32π. In this paper we give, for site percolation on the triangular lattice, a rigorous proof for the formula of E_H(n) above, and a rigorous upper bound for the prefactor of the logarithm in the formula of E_C(n).","publication_date":{"day":29,"month":3,"year":2016,"errors":{}}},"translated_abstract":"We study critical percolation on a regular planar lattice. Let E_G(n) be the expected number of open clusters intersecting or hitting the line segment [0,n]. (For the subscript G we either take H, when we restrict to the upper halfplane, or C, when we consider the full lattice). Cardy (2001) (see also Yu, Saleur and Haas (2008)) derived heuristically that E_H(n) = An + √(3)/4π(n) + o((n)), where A is some constant. Recently Kovács, Iglói and Cardy (2012) derived heuristically (as a special case of a more general formula) that a similar result holds for E_C(n) with the constant √(3)/4π replaced by 5√(3)/32π. In this paper we give, for site percolation on the triangular lattice, a rigorous proof for the formula of E_H(n) above, and a rigorous upper bound for the prefactor of the logarithm in the formula of E_C(n).","internal_url":"https://www.academia.edu/72794607/The_expected_number_of_critical_percolation_clusters_intersecting_a_line_segment","translated_internal_url":"","created_at":"2022-03-02T06:29:55.049-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":102711148,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":81579813,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/81579813/thumbnails/1.jpg","file_name":"1505.08046v2.pdf","download_url":"https://www.academia.edu/attachments/81579813/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_expected_number_of_critical_percolat.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/81579813/1505.08046v2-libre.pdf?1646231818=\u0026response-content-disposition=attachment%3B+filename%3DThe_expected_number_of_critical_percolat.pdf\u0026Expires=1732775116\u0026Signature=ShuYSqhZBSLxxzqDWp1bcz7c8g803InAg74mcTg~ychxMUh8qpItPP5iqCS-iovF2JOavdmq8EJVKM6Hv-kpKR4FrsdX9Tf7F1R3DLFNIgJZsmjI1X-WNOa0Rg0GWiholE4yrimOPksZHKF22YCIUsrSfDtr4j5eZrr-iYjz7G6Bmb58CVBdlIC-qYO5DctVZvJ-C77eHuNWYF2LN6lC0Y5xZ6uoM2hgtwpArOVEP1Ewa-pVRzKJRrMqfyWVSdp7RZ8gAeACGKcK5fd0OGkwdj5bD11TJsMqmTgbuGv4~vSwrNLCQw4BmcOmDzeQfhIRiW31NpBU926kxKl0z6RpMw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"The_expected_number_of_critical_percolation_clusters_intersecting_a_line_segment","translated_slug":"","page_count":12,"language":"en","content_type":"Work","owner":{"id":102711148,"first_name":"Rene","middle_initials":null,"last_name":"Conijn","page_name":"ReneConijn","domain_name":"independent","created_at":"2019-02-18T02:03:26.325-08:00","display_name":"Rene Conijn","url":"https://independent.academia.edu/ReneConijn"},"attachments":[{"id":81579813,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/81579813/thumbnails/1.jpg","file_name":"1505.08046v2.pdf","download_url":"https://www.academia.edu/attachments/81579813/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_expected_number_of_critical_percolat.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/81579813/1505.08046v2-libre.pdf?1646231818=\u0026response-content-disposition=attachment%3B+filename%3DThe_expected_number_of_critical_percolat.pdf\u0026Expires=1732775116\u0026Signature=ShuYSqhZBSLxxzqDWp1bcz7c8g803InAg74mcTg~ychxMUh8qpItPP5iqCS-iovF2JOavdmq8EJVKM6Hv-kpKR4FrsdX9Tf7F1R3DLFNIgJZsmjI1X-WNOa0Rg0GWiholE4yrimOPksZHKF22YCIUsrSfDtr4j5eZrr-iYjz7G6Bmb58CVBdlIC-qYO5DctVZvJ-C77eHuNWYF2LN6lC0Y5xZ6uoM2hgtwpArOVEP1Ewa-pVRzKJRrMqfyWVSdp7RZ8gAeACGKcK5fd0OGkwdj5bD11TJsMqmTgbuGv4~vSwrNLCQw4BmcOmDzeQfhIRiW31NpBU926kxKl0z6RpMw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":81579812,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/81579812/thumbnails/1.jpg","file_name":"1505.08046v2.pdf","download_url":"https://www.academia.edu/attachments/81579812/download_file","bulk_download_file_name":"The_expected_number_of_critical_percolat.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/81579812/1505.08046v2-libre.pdf?1646231817=\u0026response-content-disposition=attachment%3B+filename%3DThe_expected_number_of_critical_percolat.pdf\u0026Expires=1732775116\u0026Signature=JLTVr3~r6YWucQPuI65OK7sOXvCYt7XebE0NT0LuVM4YPHKR~WpVP2u6jAfoWsPfoLLwpXL39dNhVnyoh-U9dG2kC2jMslW1nPIE-V6RxIscruAmkw~yLUh9tzX3L3UVgxVECZDVz4wDwhYpt0iK2nxAiq7foXOOZHfpqP6ZeQlPrAOJ6RyzVj3f2za7Of2QLjdOf7lFWrqqQI6p2Ms5ZoU43TvedtkJKlMWNtf6bkQaFbETs65KZpCAovfQtG9zTZaeiR01ph3PTwwZut~xaYDUR1-OoIV~RyO3Vk54zMr7o6jK-r3ZX8fOLh5Kuz~b2kuQWW62emksGcWbklilRg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":892,"name":"Statistics","url":"https://www.academia.edu/Documents/in/Statistics"}],"urls":[{"id":18150951,"url":"https://arxiv.org/pdf/1505.08046v2.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="62924981"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/62924981/Factorization_formulas_for_2D_critical_percolation_revisited"><img alt="Research paper thumbnail of Factorization formulas for 2D critical percolation, revisited" class="work-thumbnail" src="https://attachments.academia-assets.com/75531425/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/62924981/Factorization_formulas_for_2D_critical_percolation_revisited">Factorization formulas for 2D critical percolation, revisited</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We consider critical site percolation on the triangular lattice in the upper half-plane. Let u1,u...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We consider critical site percolation on the triangular lattice in the upper half-plane. Let u1,u2 be two sites on the boundary and w a site in the interior. It was predicted by Simmons et al. (2007) that the ratio P(nu1↔nu2↔nw)2/P(nu1↔nu2)⋅P(nu1↔nw)⋅P(nu2↔nw) converges to KF as n→∞, where x↔y denotes that x and y are in the same cluster, and KF is a constant. Beliaev and Izyurov (2012) proved an analog of this in the scaling limit. We prove, using their result and a generalized coupling argument, the earlier mentioned prediction. Furthermore we prove a factorization formula for P(nu2↔[nu1,nu1+s];nw↔[nu1,nu1+s]), where s&gt;0.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="eab91ff78cfee043c3de560fc720fe8b" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":75531425,"asset_id":62924981,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/75531425/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="62924981"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="62924981"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 62924981; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=62924981]").text(description); $(".js-view-count[data-work-id=62924981]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 62924981; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='62924981']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 62924981, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "eab91ff78cfee043c3de560fc720fe8b" } } $('.js-work-strip[data-work-id=62924981]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":62924981,"title":"Factorization formulas for 2D critical percolation, revisited","translated_title":"","metadata":{"abstract":"We consider critical site percolation on the triangular lattice in the upper half-plane. Let u1,u2 be two sites on the boundary and w a site in the interior. It was predicted by Simmons et al. (2007) that the ratio P(nu1↔nu2↔nw)2/P(nu1↔nu2)⋅P(nu1↔nw)⋅P(nu2↔nw) converges to KF as n→∞, where x↔y denotes that x and y are in the same cluster, and KF is a constant. Beliaev and Izyurov (2012) proved an analog of this in the scaling limit. We prove, using their result and a generalized coupling argument, the earlier mentioned prediction. Furthermore we prove a factorization formula for P(nu2↔[nu1,nu1+s];nw↔[nu1,nu1+s]), where s\u0026gt;0.","publication_date":{"day":null,"month":null,"year":2015,"errors":{}}},"translated_abstract":"We consider critical site percolation on the triangular lattice in the upper half-plane. Let u1,u2 be two sites on the boundary and w a site in the interior. It was predicted by Simmons et al. (2007) that the ratio P(nu1↔nu2↔nw)2/P(nu1↔nu2)⋅P(nu1↔nw)⋅P(nu2↔nw) converges to KF as n→∞, where x↔y denotes that x and y are in the same cluster, and KF is a constant. Beliaev and Izyurov (2012) proved an analog of this in the scaling limit. We prove, using their result and a generalized coupling argument, the earlier mentioned prediction. Furthermore we prove a factorization formula for P(nu2↔[nu1,nu1+s];nw↔[nu1,nu1+s]), where s\u0026gt;0.","internal_url":"https://www.academia.edu/62924981/Factorization_formulas_for_2D_critical_percolation_revisited","translated_internal_url":"","created_at":"2021-12-01T22:00:50.931-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":102711148,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":75531425,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/75531425/thumbnails/1.jpg","file_name":"1502.pdf","download_url":"https://www.academia.edu/attachments/75531425/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Factorization_formulas_for_2D_critical_p.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/75531425/1502-libre.pdf?1638577793=\u0026response-content-disposition=attachment%3B+filename%3DFactorization_formulas_for_2D_critical_p.pdf\u0026Expires=1732775116\u0026Signature=E5vhUjEOux~hb5NWCcJ~u2HL7zgzkV6~oXHwSskNjage5m6ckTJB3BJiW6-m7XxRYaEJBByIogwuahuJ1XcLiztvNZxgBbwJLwAUqBxOl9Ux5KA3d5bdjI392i2nnXwzrd~0pkyaDLDH2uZUV4~VwpBJ9RTt4-SqoINUOAQW4jpI2FWHdjtCjE9SlthiSR8JbRCVRGBWdskKzWBnnDo2i9vxHuvFvsynoNgoBBRBVWE4iu-XZAKhW62EI6AgWIuHES2kdYxb6oAmEsFonASsksZ4BxNSphErlFSMXetbFLmMxoVYrNe7qL0J7s-Lz~iuzr4jJUTNpbkoyZNJaNCcxA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Factorization_formulas_for_2D_critical_percolation_revisited","translated_slug":"","page_count":17,"language":"en","content_type":"Work","owner":{"id":102711148,"first_name":"Rene","middle_initials":null,"last_name":"Conijn","page_name":"ReneConijn","domain_name":"independent","created_at":"2019-02-18T02:03:26.325-08:00","display_name":"Rene Conijn","url":"https://independent.academia.edu/ReneConijn"},"attachments":[{"id":75531425,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/75531425/thumbnails/1.jpg","file_name":"1502.pdf","download_url":"https://www.academia.edu/attachments/75531425/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Factorization_formulas_for_2D_critical_p.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/75531425/1502-libre.pdf?1638577793=\u0026response-content-disposition=attachment%3B+filename%3DFactorization_formulas_for_2D_critical_p.pdf\u0026Expires=1732775116\u0026Signature=E5vhUjEOux~hb5NWCcJ~u2HL7zgzkV6~oXHwSskNjage5m6ckTJB3BJiW6-m7XxRYaEJBByIogwuahuJ1XcLiztvNZxgBbwJLwAUqBxOl9Ux5KA3d5bdjI392i2nnXwzrd~0pkyaDLDH2uZUV4~VwpBJ9RTt4-SqoINUOAQW4jpI2FWHdjtCjE9SlthiSR8JbRCVRGBWdskKzWBnnDo2i9vxHuvFvsynoNgoBBRBVWE4iu-XZAKhW62EI6AgWIuHES2kdYxb6oAmEsFonASsksZ4BxNSphErlFSMXetbFLmMxoVYrNe7qL0J7s-Lz~iuzr4jJUTNpbkoyZNJaNCcxA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":75531424,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/75531424/thumbnails/1.jpg","file_name":"1502.pdf","download_url":"https://www.academia.edu/attachments/75531424/download_file","bulk_download_file_name":"Factorization_formulas_for_2D_critical_p.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/75531424/1502-libre.pdf?1638577794=\u0026response-content-disposition=attachment%3B+filename%3DFactorization_formulas_for_2D_critical_p.pdf\u0026Expires=1732775116\u0026Signature=WuUFBQhpupNyJcPwDs4OZMiz9IktV19G4FeBBh8KOjZjVNf9z5ZujTZYt1Me36RjKRI4AYj0YWYWBq~1Vih4P~j1EGyiFwZcJ4bhePvtD07s03MKqzZBbtPheKMRT2QfxwgmYoJbhA2-zW57Af7kQshFGm51luLjoIXr3pQ6L~6fVxPRHKWcp6AmAnBTDw8uL3Mlwi~Qe2iw~F9MLcAtjWWcvlbqJRQRLlCmFg5OQJrjQQXrDwJG5fNVWYujuGYaHjhHnCx9Q5l8ZVjqIODsQYo3JQ~7kdMXymdrEkwKqKgDFBBqHHla~na-2fTz6QIRHFqNNsE1N4Grxwo3tF039w__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":892,"name":"Statistics","url":"https://www.academia.edu/Documents/in/Statistics"},{"id":3079415,"name":"Finance and Investment Banking","url":"https://www.academia.edu/Documents/in/Finance_and_Investment_Banking"}],"urls":[{"id":14681358,"url":"http://export.arxiv.org/pdf/1502.04387"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="54335823"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/54335823/Factorization_Formulas_for_2D_Critical_Percolation_Revisited"><img alt="Research paper thumbnail of Factorization Formulas for $2D$ Critical Percolation, Revisited" class="work-thumbnail" src="https://attachments.academia-assets.com/70749716/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/54335823/Factorization_Formulas_for_2D_Critical_Percolation_Revisited">Factorization Formulas for $2D$ Critical Percolation, Revisited</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We consider critical site percolation on the triangular lattice in the upper half-plane. Let $u_1...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We consider critical site percolation on the triangular lattice in the upper half-plane. Let $u_1, u_2$ be two sites on the boundary and $w$ a site in the interior of the half-plane. It was predicted by Simmons, Kleban and Ziff in a paper from 2007 that the ratio $\mathbb{P}(nu_1 \leftrightarrow nu_2 \leftrightarrow nw)^{2}\,/\,\mathbb{P}(nu_1 \leftrightarrow nu_2)\cdot\mathbb{P}(nu_1 \leftrightarrow nw)\cdot\mathbb{P}(nu_2 \leftrightarrow nw)$ converges to $K_F$ as $n \to \infty$, where $x\leftrightarrow y$ denotes the event that $x$ and $y$ are in the same open cluster, and $K_F$ is an explicitly known constant. Beliaev and Izyurov proved in a paper in 2012 an analog of this factorization in the scaling limit. We prove, using their result and a generalized coupling argument, the earlier mentioned prediction. Furthermore we prove a factorization formula for the probability $\mathbb{P}(nu_2 \leftrightarrow [nu_1,nu_1+s];\, nw \leftrightarrow [nu_1,nu_1+s])$, where $s>0$.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="81cd6188f41efe1529fba52ad1898a4b" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":70749716,"asset_id":54335823,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/70749716/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="54335823"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="54335823"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 54335823; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=54335823]").text(description); $(".js-view-count[data-work-id=54335823]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 54335823; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='54335823']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 54335823, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "81cd6188f41efe1529fba52ad1898a4b" } } $('.js-work-strip[data-work-id=54335823]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":54335823,"title":"Factorization Formulas for $2D$ Critical Percolation, Revisited","translated_title":"","metadata":{"abstract":"We consider critical site percolation on the triangular lattice in the upper half-plane. Let $u_1, u_2$ be two sites on the boundary and $w$ a site in the interior of the half-plane. It was predicted by Simmons, Kleban and Ziff in a paper from 2007 that the ratio $\\mathbb{P}(nu_1 \\leftrightarrow nu_2 \\leftrightarrow nw)^{2}\\,/\\,\\mathbb{P}(nu_1 \\leftrightarrow nu_2)\\cdot\\mathbb{P}(nu_1 \\leftrightarrow nw)\\cdot\\mathbb{P}(nu_2 \\leftrightarrow nw)$ converges to $K_F$ as $n \\to \\infty$, where $x\\leftrightarrow y$ denotes the event that $x$ and $y$ are in the same open cluster, and $K_F$ is an explicitly known constant. Beliaev and Izyurov proved in a paper in 2012 an analog of this factorization in the scaling limit. We prove, using their result and a generalized coupling argument, the earlier mentioned prediction. Furthermore we prove a factorization formula for the probability $\\mathbb{P}(nu_2 \\leftrightarrow [nu_1,nu_1+s];\\, nw \\leftrightarrow [nu_1,nu_1+s])$, where $s\u003e0$.","publication_date":{"day":15,"month":2,"year":2015,"errors":{}}},"translated_abstract":"We consider critical site percolation on the triangular lattice in the upper half-plane. Let $u_1, u_2$ be two sites on the boundary and $w$ a site in the interior of the half-plane. It was predicted by Simmons, Kleban and Ziff in a paper from 2007 that the ratio $\\mathbb{P}(nu_1 \\leftrightarrow nu_2 \\leftrightarrow nw)^{2}\\,/\\,\\mathbb{P}(nu_1 \\leftrightarrow nu_2)\\cdot\\mathbb{P}(nu_1 \\leftrightarrow nw)\\cdot\\mathbb{P}(nu_2 \\leftrightarrow nw)$ converges to $K_F$ as $n \\to \\infty$, where $x\\leftrightarrow y$ denotes the event that $x$ and $y$ are in the same open cluster, and $K_F$ is an explicitly known constant. Beliaev and Izyurov proved in a paper in 2012 an analog of this factorization in the scaling limit. We prove, using their result and a generalized coupling argument, the earlier mentioned prediction. Furthermore we prove a factorization formula for the probability $\\mathbb{P}(nu_2 \\leftrightarrow [nu_1,nu_1+s];\\, nw \\leftrightarrow [nu_1,nu_1+s])$, where $s\u003e0$.","internal_url":"https://www.academia.edu/54335823/Factorization_Formulas_for_2D_Critical_Percolation_Revisited","translated_internal_url":"","created_at":"2021-09-30T07:52:09.502-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":102711148,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":70749716,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/70749716/thumbnails/1.jpg","file_name":"1502.04387.pdf","download_url":"https://www.academia.edu/attachments/70749716/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Factorization_Formulas_for_2D_Critical_P.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/70749716/1502.04387-libre.pdf?1633018236=\u0026response-content-disposition=attachment%3B+filename%3DFactorization_Formulas_for_2D_Critical_P.pdf\u0026Expires=1732775116\u0026Signature=PJDjbscYVMc0hKiq801qggTtW5cj5SDWMzm9hYs9SDmzvALbhmlKiLCZnD5yZlR~FxzBK9jMM42jHbofw~A-mKgZtsFrkiLbK6Pm3oMrvO4rCyz95BqtHTRIS0HGirlI0ut6FSs6mz~4NWt2NZG6fyOyxiCJac82QprgvkIZ1hjvUynzydEdpjL0HShipSFbzKx83G2-uv0r8CuWyBKYNZETEll8pFPQ1ganwJPCmecZChSTe8msFRNSfaZbpjc7cpE1IOzc4Sr0hhK8TInaBAt-kPFZdBKg-6r~LY~f9HLHVY9TW~VoJwFnsDde5hP5F7kh2aZc6nrAfI7j3~Vq1A__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Factorization_Formulas_for_2D_Critical_Percolation_Revisited","translated_slug":"","page_count":17,"language":"en","content_type":"Work","owner":{"id":102711148,"first_name":"Rene","middle_initials":null,"last_name":"Conijn","page_name":"ReneConijn","domain_name":"independent","created_at":"2019-02-18T02:03:26.325-08:00","display_name":"Rene Conijn","url":"https://independent.academia.edu/ReneConijn"},"attachments":[{"id":70749716,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/70749716/thumbnails/1.jpg","file_name":"1502.04387.pdf","download_url":"https://www.academia.edu/attachments/70749716/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Factorization_Formulas_for_2D_Critical_P.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/70749716/1502.04387-libre.pdf?1633018236=\u0026response-content-disposition=attachment%3B+filename%3DFactorization_Formulas_for_2D_Critical_P.pdf\u0026Expires=1732775116\u0026Signature=PJDjbscYVMc0hKiq801qggTtW5cj5SDWMzm9hYs9SDmzvALbhmlKiLCZnD5yZlR~FxzBK9jMM42jHbofw~A-mKgZtsFrkiLbK6Pm3oMrvO4rCyz95BqtHTRIS0HGirlI0ut6FSs6mz~4NWt2NZG6fyOyxiCJac82QprgvkIZ1hjvUynzydEdpjL0HShipSFbzKx83G2-uv0r8CuWyBKYNZETEll8pFPQ1ganwJPCmecZChSTe8msFRNSfaZbpjc7cpE1IOzc4Sr0hhK8TInaBAt-kPFZdBKg-6r~LY~f9HLHVY9TW~VoJwFnsDde5hP5F7kh2aZc6nrAfI7j3~Vq1A__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[],"urls":[{"id":11823191,"url":"http://arxiv.org/abs/1502.04387"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="54335806"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/54335806/The_expected_number_of_critical_percolation_clusters_intersecting_a_line_segment"><img alt="Research paper thumbnail of The expected number of critical percolation clusters intersecting a line segment" class="work-thumbnail" src="https://attachments.academia-assets.com/70749702/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/54335806/The_expected_number_of_critical_percolation_clusters_intersecting_a_line_segment">The expected number of critical percolation clusters intersecting a line segment</a></div><div class="wp-workCard_item"><span>Electronic Communications in Probability</span><span>, 2016</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2b50175132cb4b56941911e440c47f0e" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":70749702,"asset_id":54335806,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/70749702/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="54335806"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="54335806"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 54335806; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=54335806]").text(description); $(".js-view-count[data-work-id=54335806]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 54335806; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='54335806']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 54335806, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "2b50175132cb4b56941911e440c47f0e" } } $('.js-work-strip[data-work-id=54335806]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":54335806,"title":"The expected number of critical percolation clusters intersecting a line segment","translated_title":"","metadata":{"publisher":"Institute of Mathematical Statistics","grobid_abstract":"We study critical percolation on a regular planar lattice. Let E G (n) be the expected number of open clusters intersecting or hitting the line segment [0, n]. (For the subscript G we either take H, when we restrict to the upper halfplane, or C, when we consider the full lattice). Cardy [Car01] (see also Yu, Saleur and Haas [YSH08]) derived heuristically that E H (n) = An + √ 3 4π log(n) + o(log(n)), where A is some constant. Recently Kovács, Iglói and Cardy derived in [KIC12] heuristically (as a special case of a more general formula) that a similar result holds for E C (n) with the constant √ 3 4π replaced by 5 √ 3 32π. In this paper we give, for site percolation on the triangular lattice, a rigorous proof for the formula of E H (n) above, and a rigorous upper bound for the prefactor of the logarithm in the formula of E C (n).","publication_date":{"day":null,"month":null,"year":2016,"errors":{}},"publication_name":"Electronic Communications in Probability","grobid_abstract_attachment_id":70749702},"translated_abstract":null,"internal_url":"https://www.academia.edu/54335806/The_expected_number_of_critical_percolation_clusters_intersecting_a_line_segment","translated_internal_url":"","created_at":"2021-09-30T07:52:07.499-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":102711148,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":70749702,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/70749702/thumbnails/1.jpg","file_name":"1505.pdf","download_url":"https://www.academia.edu/attachments/70749702/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_expected_number_of_critical_percolat.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/70749702/1505-libre.pdf?1633018235=\u0026response-content-disposition=attachment%3B+filename%3DThe_expected_number_of_critical_percolat.pdf\u0026Expires=1732775116\u0026Signature=Wr6rDiGYA8AXhiFvRbMUg4IH-IAkhEi8kx1ff1e7LXs92XdA6wAoBbLEllpRGVsD3lZ79upMMiULqlIuyztT1TLem0AubSgHtvFgttrwy9LEpGdw9pk7A8oCrfMZlNaqLH2bzZHAaABKnaawwY5emWa56M4oA-J0DasAH-K9~MYSoZMgs3z5AhCCzMo3W81pFZfaTUVo3iTRV72xdVEL2eUos8JbMapLUturcu3IsieIBk6RETd6uw0alUiGr6P1PEhawGimDvgKRpqOtTXQVxgxQ~p5Rlhu8npTI7-3LCOmjYHzRsFBY~~SoIYHYERgDvCi4G5Vv~ikocduo7oXcQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"The_expected_number_of_critical_percolation_clusters_intersecting_a_line_segment","translated_slug":"","page_count":12,"language":"en","content_type":"Work","owner":{"id":102711148,"first_name":"Rene","middle_initials":null,"last_name":"Conijn","page_name":"ReneConijn","domain_name":"independent","created_at":"2019-02-18T02:03:26.325-08:00","display_name":"Rene Conijn","url":"https://independent.academia.edu/ReneConijn"},"attachments":[{"id":70749702,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/70749702/thumbnails/1.jpg","file_name":"1505.pdf","download_url":"https://www.academia.edu/attachments/70749702/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNiw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_expected_number_of_critical_percolat.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/70749702/1505-libre.pdf?1633018235=\u0026response-content-disposition=attachment%3B+filename%3DThe_expected_number_of_critical_percolat.pdf\u0026Expires=1732775116\u0026Signature=Wr6rDiGYA8AXhiFvRbMUg4IH-IAkhEi8kx1ff1e7LXs92XdA6wAoBbLEllpRGVsD3lZ79upMMiULqlIuyztT1TLem0AubSgHtvFgttrwy9LEpGdw9pk7A8oCrfMZlNaqLH2bzZHAaABKnaawwY5emWa56M4oA-J0DasAH-K9~MYSoZMgs3z5AhCCzMo3W81pFZfaTUVo3iTRV72xdVEL2eUos8JbMapLUturcu3IsieIBk6RETd6uw0alUiGr6P1PEhawGimDvgKRpqOtTXQVxgxQ~p5Rlhu8npTI7-3LCOmjYHzRsFBY~~SoIYHYERgDvCi4G5Vv~ikocduo7oXcQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":892,"name":"Statistics","url":"https://www.academia.edu/Documents/in/Statistics"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="54335787"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/54335787/Conformal_Measure_Ensembles_for_Percolation_and_the_FK_Ising_model"><img alt="Research paper thumbnail of Conformal Measure Ensembles for Percolation and the FK-Ising model" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/54335787/Conformal_Measure_Ensembles_for_Percolation_and_the_FK_Ising_model">Conformal Measure Ensembles for Percolation and the FK-Ising model</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Under some general assumptions we construct the scaling limit of open clusters and their associat...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Under some general assumptions we construct the scaling limit of open clusters and their associated counting measures in a class of two dimensional percolation models. Our results apply, in particular, to critical Bernoulli site percolation on the triangular lattice. We also provide conditional results for the critical FK-Ising model on the square lattice. Fundamental properties of the scaling limit, such as conformal covariance, are explored. Applications such as the scaling limit of the largest cluster in a bounded domain and a geometric representation of the magnetization field for the critical Ising model are presented.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="54335787"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="54335787"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 54335787; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=54335787]").text(description); $(".js-view-count[data-work-id=54335787]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 54335787; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='54335787']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 54335787, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=54335787]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":54335787,"title":"Conformal Measure Ensembles for Percolation and the FK-Ising model","translated_title":"","metadata":{"abstract":"Under some general assumptions we construct the scaling limit of open clusters and their associated counting measures in a class of two dimensional percolation models. Our results apply, in particular, to critical Bernoulli site percolation on the triangular lattice. We also provide conditional results for the critical FK-Ising model on the square lattice. Fundamental properties of the scaling limit, such as conformal covariance, are explored. Applications such as the scaling limit of the largest cluster in a bounded domain and a geometric representation of the magnetization field for the critical Ising model are presented."},"translated_abstract":"Under some general assumptions we construct the scaling limit of open clusters and their associated counting measures in a class of two dimensional percolation models. Our results apply, in particular, to critical Bernoulli site percolation on the triangular lattice. We also provide conditional results for the critical FK-Ising model on the square lattice. Fundamental properties of the scaling limit, such as conformal covariance, are explored. Applications such as the scaling limit of the largest cluster in a bounded domain and a geometric representation of the magnetization field for the critical Ising model are presented.","internal_url":"https://www.academia.edu/54335787/Conformal_Measure_Ensembles_for_Percolation_and_the_FK_Ising_model","translated_internal_url":"","created_at":"2021-09-30T07:52:05.657-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":102711148,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[],"slug":"Conformal_Measure_Ensembles_for_Percolation_and_the_FK_Ising_model","translated_slug":"","page_count":null,"language":"en","content_type":"Work","owner":{"id":102711148,"first_name":"Rene","middle_initials":null,"last_name":"Conijn","page_name":"ReneConijn","domain_name":"independent","created_at":"2019-02-18T02:03:26.325-08:00","display_name":"Rene Conijn","url":"https://independent.academia.edu/ReneConijn"},"attachments":[],"research_interests":[],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="54335771"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/54335771/On_the_size_of_the_largest_cluster_in_2D_critical_percolation"><img alt="Research paper thumbnail of On the size of the largest cluster in 2D critical percolation" class="work-thumbnail" src="https://attachments.academia-assets.com/70749697/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/54335771/On_the_size_of_the_largest_cluster_in_2D_critical_percolation">On the size of the largest cluster in 2D critical percolation</a></div><div class="wp-workCard_item"><span>Electronic Communications in Probability</span><span>, 2012</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="d47c86c6c6ab6a5adc5eef65f7b08a2a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":70749697,"asset_id":54335771,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/70749697/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="54335771"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="54335771"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 54335771; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=54335771]").text(description); $(".js-view-count[data-work-id=54335771]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 54335771; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='54335771']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 54335771, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "d47c86c6c6ab6a5adc5eef65f7b08a2a" } } $('.js-work-strip[data-work-id=54335771]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":54335771,"title":"On the size of the largest cluster in 2D critical percolation","translated_title":"","metadata":{"publisher":"Institute of Mathematical Statistics","grobid_abstract":"We consider (near-)critical percolation on the square lattice. Let Mn be the size of the largest open cluster contained in the box [−n, n] 2 , and let π(n) be the probability that there is an open path from O to the boundary of the box. It is well-known (see [BCKS01]) that for all 0 \u003c a \u003c b the probability that Mn is smaller than an 2 π(n) and the probability that Mn is larger than bn 2 π(n) are bounded away from 0 as n → ∞. It is a natural question, which arises for instance in the study of so-called frozenpercolation processes, if a similar result holds for the probability that Mn is between an 2 π(n) and bn 2 π(n). By a suitable partition of the box, and a careful construction involving the building blocks, we show that the answer to this question is affirmative. The 'sublinearity' of 1/π(n) appears to be essential for the argument.","publication_date":{"day":null,"month":null,"year":2012,"errors":{}},"publication_name":"Electronic Communications in Probability","grobid_abstract_attachment_id":70749697},"translated_abstract":null,"internal_url":"https://www.academia.edu/54335771/On_the_size_of_the_largest_cluster_in_2D_critical_percolation","translated_internal_url":"","created_at":"2021-09-30T07:52:02.781-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":102711148,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":70749697,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/70749697/thumbnails/1.jpg","file_name":"1208.pdf","download_url":"https://www.academia.edu/attachments/70749697/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_the_size_of_the_largest_cluster_in_2D.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/70749697/1208-libre.pdf?1633018237=\u0026response-content-disposition=attachment%3B+filename%3DOn_the_size_of_the_largest_cluster_in_2D.pdf\u0026Expires=1732775116\u0026Signature=cEMirdpdJsfO3opjeEu6pZDXFX0xLuCPxIpKciWPiuWot4dZE8cBECgr32nB63G8GCN8uG1mo9cbcoTrQb28vAdignbD5m0E~NTsygH6l3polLuZvIqyZBdvdhcIlU5wBIqAx~HGvc8jY0ihJhshKs~bRE3b5tmKp2Ra5bGvHW9nKun34NkYMUwJeLvVXvhcaY-wIqfae9jUG8S7H2BkugeBeTKK4HE9avrqN7MaU0uE12i2g4uAMyEYF5wXbMjqFgoyOxcey77FdGJvugeBr5cd-sPZZ55EjYuwwi01sSevuii~2~a4AVkxGuOWGQNd4Rc146DxgzN8n96w3rSigw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"On_the_size_of_the_largest_cluster_in_2D_critical_percolation","translated_slug":"","page_count":12,"language":"en","content_type":"Work","owner":{"id":102711148,"first_name":"Rene","middle_initials":null,"last_name":"Conijn","page_name":"ReneConijn","domain_name":"independent","created_at":"2019-02-18T02:03:26.325-08:00","display_name":"Rene Conijn","url":"https://independent.academia.edu/ReneConijn"},"attachments":[{"id":70749697,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/70749697/thumbnails/1.jpg","file_name":"1208.pdf","download_url":"https://www.academia.edu/attachments/70749697/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_the_size_of_the_largest_cluster_in_2D.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/70749697/1208-libre.pdf?1633018237=\u0026response-content-disposition=attachment%3B+filename%3DOn_the_size_of_the_largest_cluster_in_2D.pdf\u0026Expires=1732775117\u0026Signature=aAgW1Dvjwb8RV2hvn6gk8b3i9sD2Zl2l7aFFB8HjIXVEaC0FHPaoKGl~RKDVnDQXQ23lVs3vA4YlL9fxJxfp5ivK06cgy-gVCXsWpXHCfoMcOPS2xjuR1zHE6LdcseyTpEkq2xoDc3W1yexyqCDOwRHaVBbCJ9f5BdS4nyyIJYXkP7XJWVOc96qWktbYqGDA2pNCNIi89om2ynEv1k5qaVBC9~sqeDOdP6~sbhbkdIfFHA3QhIxaarqfSj8MG3tT1v4R-XOuTwsy~ZOPWDfN5a39KW3AO8GEm1Fu7gA6HHDvweXYHbENnc1xjq0t0T3t137jtXgO4nBX8IU6WFTXow__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":892,"name":"Statistics","url":"https://www.academia.edu/Documents/in/Statistics"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="54335511"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/54335511/The_gaps_between_the_sizes_of_large_clusters_in_2D_critical_percolation"><img alt="Research paper thumbnail of The gaps between the sizes of large clusters in 2D critical percolation" class="work-thumbnail" src="https://attachments.academia-assets.com/70749541/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/54335511/The_gaps_between_the_sizes_of_large_clusters_in_2D_critical_percolation">The gaps between the sizes of large clusters in 2D critical percolation</a></div><div class="wp-workCard_item"><span>Electronic Communications in Probability</span><span>, 2013</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="207226ba4b39c269285d728ad9f61a5a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":70749541,"asset_id":54335511,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/70749541/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="54335511"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="54335511"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 54335511; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=54335511]").text(description); $(".js-view-count[data-work-id=54335511]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 54335511; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='54335511']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 54335511, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "207226ba4b39c269285d728ad9f61a5a" } } $('.js-work-strip[data-work-id=54335511]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":54335511,"title":"The gaps between the sizes of large clusters in 2D critical percolation","translated_title":"","metadata":{"publisher":"Institute of Mathematical Statistics","grobid_abstract":"Consider critical bond percolation on a large 2n × 2n box on the square lattice. It is well-known that the size (i.e. number of vertices) of the largest open cluster is, with high probability, of order n 2 π(n), where π(n) denotes the probability that there is an open path from the center to the boundary of the box. The same result holds for the second-largest cluster, the third largest cluster etcetera. Járai showed that the differences between the sizes of these clusters is, with high probability, at least of order n 2 π(n). Although this bound was enough for his applications (to incipient infinite clusters), he believed, but had no proof, that the differences are in fact of the same order as the cluster sizes themselves, i.e. n 2 π(n). Our main result is a proof that this is indeed the case.","publication_date":{"day":null,"month":null,"year":2013,"errors":{}},"publication_name":"Electronic Communications in Probability","grobid_abstract_attachment_id":70749541},"translated_abstract":null,"internal_url":"https://www.academia.edu/54335511/The_gaps_between_the_sizes_of_large_clusters_in_2D_critical_percolation","translated_internal_url":"","created_at":"2021-09-30T07:51:20.013-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":102711148,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":70749541,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/70749541/thumbnails/1.jpg","file_name":"21868D.pdf","download_url":"https://www.academia.edu/attachments/70749541/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_gaps_between_the_sizes_of_large_clus.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/70749541/21868D-libre.pdf?1633018243=\u0026response-content-disposition=attachment%3B+filename%3DThe_gaps_between_the_sizes_of_large_clus.pdf\u0026Expires=1732775117\u0026Signature=WdJxNLVCQs-hC-y0t3~K5TMoGQQ5TUg2w99IIg-XvcRJovW4uN~RP1eCBXRa4Flx8bx6XJ0KwCdXg0ZZFd53JyJaS7AT5J7Kp1r6WWtSDgB5RITXcJ92AhP2EOIIkP4z7CcN3LfWbMQ2drwnmTOw7dnaIH2QVg0QoUC2FnsnGjvFvOHyglNLyUJFpid97oejS78wvAQhmD1Zq0uqkfL-152SV2FKsgfdKDIHacVAJAYplRWxlewHYJW0OEsuC3S96W-KbyO~fy1eFIss7GriRXqjvx~akQxJBTKmuImYwp4IUk9QLknrQaeFWlkgbUNjKKGjL-AbQSyIEgRl7hQ3SA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"The_gaps_between_the_sizes_of_large_clusters_in_2D_critical_percolation","translated_slug":"","page_count":10,"language":"en","content_type":"Work","owner":{"id":102711148,"first_name":"Rene","middle_initials":null,"last_name":"Conijn","page_name":"ReneConijn","domain_name":"independent","created_at":"2019-02-18T02:03:26.325-08:00","display_name":"Rene Conijn","url":"https://independent.academia.edu/ReneConijn"},"attachments":[{"id":70749541,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/70749541/thumbnails/1.jpg","file_name":"21868D.pdf","download_url":"https://www.academia.edu/attachments/70749541/download_file?st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&st=MTczMjc3MTUxNyw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_gaps_between_the_sizes_of_large_clus.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/70749541/21868D-libre.pdf?1633018243=\u0026response-content-disposition=attachment%3B+filename%3DThe_gaps_between_the_sizes_of_large_clus.pdf\u0026Expires=1732775117\u0026Signature=WdJxNLVCQs-hC-y0t3~K5TMoGQQ5TUg2w99IIg-XvcRJovW4uN~RP1eCBXRa4Flx8bx6XJ0KwCdXg0ZZFd53JyJaS7AT5J7Kp1r6WWtSDgB5RITXcJ92AhP2EOIIkP4z7CcN3LfWbMQ2drwnmTOw7dnaIH2QVg0QoUC2FnsnGjvFvOHyglNLyUJFpid97oejS78wvAQhmD1Zq0uqkfL-152SV2FKsgfdKDIHacVAJAYplRWxlewHYJW0OEsuC3S96W-KbyO~fy1eFIss7GriRXqjvx~akQxJBTKmuImYwp4IUk9QLknrQaeFWlkgbUNjKKGjL-AbQSyIEgRl7hQ3SA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":892,"name":"Statistics","url":"https://www.academia.edu/Documents/in/Statistics"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> </div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/google_contacts-0dfb882d836b94dbcb4a2d123d6933fc9533eda5be911641f20b4eb428429600.js"], function() { // from javascript_helper.rb $('.js-google-connect-button').click(function(e) { e.preventDefault(); GoogleContacts.authorize_and_show_contacts(); Aedu.Dismissibles.recordClickthrough("WowProfileImportContactsPrompt"); }); $('.js-update-biography-button').click(function(e) { e.preventDefault(); Aedu.Dismissibles.recordClickthrough("UpdateUserBiographyPrompt"); $.ajax({ url: $r.api_v0_profiles_update_about_path({ subdomain_param: 'api', about: "", }), type: 'PUT', success: function(response) { location.reload(); } }); }); $('.js-work-creator-button').click(function (e) { e.preventDefault(); window.location = $r.upload_funnel_document_path({ source: encodeURIComponent(""), }); }); $('.js-video-upload-button').click(function (e) { e.preventDefault(); window.location = $r.upload_funnel_video_path({ source: encodeURIComponent(""), }); }); $('.js-do-this-later-button').click(function() { $(this).closest('.js-profile-nag-panel').remove(); Aedu.Dismissibles.recordDismissal("WowProfileImportContactsPrompt"); }); $('.js-update-biography-do-this-later-button').click(function(){ $(this).closest('.js-profile-nag-panel').remove(); Aedu.Dismissibles.recordDismissal("UpdateUserBiographyPrompt"); }); $('.wow-profile-mentions-upsell--close').click(function(){ $('.wow-profile-mentions-upsell--panel').hide(); Aedu.Dismissibles.recordDismissal("WowProfileMentionsUpsell"); }); $('.wow-profile-mentions-upsell--button').click(function(){ Aedu.Dismissibles.recordClickthrough("WowProfileMentionsUpsell"); }); new WowProfile.SocialRedesignUserWorks({ initialWorksOffset: 20, allWorksOffset: 20, maxSections: 1 }) }); </script> </div></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile_edit-5ea339ee107c863779f560dd7275595239fed73f1a13d279d2b599a28c0ecd33.js","https://a.academia-assets.com/assets/add_coauthor-22174b608f9cb871d03443cafa7feac496fb50d7df2d66a53f5ee3c04ba67f53.js","https://a.academia-assets.com/assets/tab-dcac0130902f0cc2d8cb403714dd47454f11fc6fb0e99ae6a0827b06613abc20.js","https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js"], function() { // from javascript_helper.rb window.ae = window.ae || {}; window.ae.WowProfile = window.ae.WowProfile || {}; if(Aedu.User.current && Aedu.User.current.id === $viewedUser.id) { window.ae.WowProfile.current_user_edit = {}; new WowProfileEdit.EditUploadView({ el: '.js-edit-upload-button-wrapper', model: window.$current_user, }); new AddCoauthor.AddCoauthorsController(); } var userInfoView = new WowProfile.SocialRedesignUserInfo({ recaptcha_key: "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB" }); WowProfile.router = new WowProfile.Router({ userInfoView: userInfoView }); Backbone.history.start({ pushState: true, root: "/" + $viewedUser.page_name }); new WowProfile.UserWorksNav() }); </script> </div> <div class="bootstrap login"><div class="modal fade login-modal" id="login-modal"><div class="login-modal-dialog modal-dialog"><div class="modal-content"><div class="modal-header"><button class="close close" data-dismiss="modal" type="button"><span aria-hidden="true">×</span><span class="sr-only">Close</span></button><h4 class="modal-title text-center"><strong>Log In</strong></h4></div><div class="modal-body"><div class="row"><div class="col-xs-10 col-xs-offset-1"><button class="btn btn-fb btn-lg btn-block btn-v-center-content" id="login-facebook-oauth-button"><svg style="float: left; width: 19px; line-height: 1em; margin-right: .3em;" aria-hidden="true" focusable="false" data-prefix="fab" data-icon="facebook-square" class="svg-inline--fa fa-facebook-square fa-w-14" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512"><path fill="currentColor" d="M400 32H48A48 48 0 0 0 0 80v352a48 48 0 0 0 48 48h137.25V327.69h-63V256h63v-54.64c0-62.15 37-96.48 93.67-96.48 27.14 0 55.52 4.84 55.52 4.84v61h-31.27c-30.81 0-40.42 19.12-40.42 38.73V256h68.78l-11 71.69h-57.78V480H400a48 48 0 0 0 48-48V80a48 48 0 0 0-48-48z"></path></svg><small><strong>Log in</strong> with <strong>Facebook</strong></small></button><br /><button class="btn btn-google btn-lg btn-block btn-v-center-content" id="login-google-oauth-button"><svg style="float: left; width: 22px; line-height: 1em; margin-right: .3em;" aria-hidden="true" focusable="false" data-prefix="fab" data-icon="google-plus" class="svg-inline--fa fa-google-plus fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M256,8C119.1,8,8,119.1,8,256S119.1,504,256,504,504,392.9,504,256,392.9,8,256,8ZM185.3,380a124,124,0,0,1,0-248c31.3,0,60.1,11,83,32.3l-33.6,32.6c-13.2-12.9-31.3-19.1-49.4-19.1-42.9,0-77.2,35.5-77.2,78.1S142.3,334,185.3,334c32.6,0,64.9-19.1,70.1-53.3H185.3V238.1H302.2a109.2,109.2,0,0,1,1.9,20.7c0,70.8-47.5,121.2-118.8,121.2ZM415.5,273.8v35.5H380V273.8H344.5V238.3H380V202.8h35.5v35.5h35.2v35.5Z"></path></svg><small><strong>Log in</strong> with <strong>Google</strong></small></button><br /><style type="text/css">.sign-in-with-apple-button { width: 100%; height: 52px; border-radius: 3px; border: 1px solid black; cursor: pointer; }</style><script src="https://appleid.cdn-apple.com/appleauth/static/jsapi/appleid/1/en_US/appleid.auth.js" type="text/javascript"></script><div class="sign-in-with-apple-button" data-border="false" data-color="white" id="appleid-signin"><span   ="Sign Up with Apple" class="u-fs11"></span></div><script>AppleID.auth.init({ clientId: 'edu.academia.applesignon', scope: 'name email', redirectURI: 'https://www.academia.edu/sessions', state: "7467321db7534db7dde7fdda7060d3b83ab100fde35806abc03f0301101a619f", });</script><script>// Hacky way of checking if on fast loswp if (window.loswp == null) { (function() { const Google = window?.Aedu?.Auth?.OauthButton?.Login?.Google; const Facebook = window?.Aedu?.Auth?.OauthButton?.Login?.Facebook; if (Google) { new Google({ el: '#login-google-oauth-button', rememberMeCheckboxId: 'remember_me', track: null }); } if (Facebook) { new Facebook({ el: '#login-facebook-oauth-button', rememberMeCheckboxId: 'remember_me', track: null }); } })(); }</script></div></div></div><div class="modal-body"><div class="row"><div class="col-xs-10 col-xs-offset-1"><div class="hr-heading login-hr-heading"><span class="hr-heading-text">or</span></div></div></div></div><div class="modal-body"><div class="row"><div class="col-xs-10 col-xs-offset-1"><form class="js-login-form" action="https://www.academia.edu/sessions" accept-charset="UTF-8" method="post"><input name="utf8" type="hidden" value="✓" autocomplete="off" /><input type="hidden" name="authenticity_token" value="4f5ofjlqyiKNUq8X86X1sPQsNYvAK+Ept6CyOFAlWtB5fbtBJCr4O+1u6P03+9QTVW1V52nI4g+Jj511Nld7UQ==" autocomplete="off" /><div class="form-group"><label class="control-label" for="login-modal-email-input" style="font-size: 14px;">Email</label><input class="form-control" id="login-modal-email-input" name="login" type="email" /></div><div class="form-group"><label class="control-label" for="login-modal-password-input" style="font-size: 14px;">Password</label><input class="form-control" id="login-modal-password-input" name="password" type="password" /></div><input type="hidden" name="post_login_redirect_url" id="post_login_redirect_url" value="https://independent.academia.edu/ReneConijn" autocomplete="off" /><div class="checkbox"><label><input type="checkbox" name="remember_me" id="remember_me" value="1" checked="checked" /><small style="font-size: 12px; margin-top: 2px; display: inline-block;">Remember me on this computer</small></label></div><br><input type="submit" name="commit" value="Log In" class="btn btn-primary btn-block btn-lg js-login-submit" data-disable-with="Log In" /></br></form><script>typeof window?.Aedu?.recaptchaManagedForm === 'function' && window.Aedu.recaptchaManagedForm( document.querySelector('.js-login-form'), document.querySelector('.js-login-submit') );</script><small style="font-size: 12px;"><br />or <a data-target="#login-modal-reset-password-container" data-toggle="collapse" href="javascript:void(0)">reset password</a></small><div class="collapse" id="login-modal-reset-password-container"><br /><div class="well margin-0x"><form class="js-password-reset-form" action="https://www.academia.edu/reset_password" accept-charset="UTF-8" method="post"><input name="utf8" type="hidden" value="✓" autocomplete="off" /><input type="hidden" name="authenticity_token" value="IDhvY8+NF6Ux+WDYLXCr4+OYLfaw8RG5E4EIQpmMizq4u7xc0s0lvFHFJzLpLopAQtlNmhkSEp8tricP//6quw==" autocomplete="off" /><p>Enter the email address you signed up with and we'll email you a reset link.</p><div class="form-group"><input class="form-control" name="email" type="email" /></div><script src="https://recaptcha.net/recaptcha/api.js" async defer></script> <script> var invisibleRecaptchaSubmit = function () { var closestForm = function (ele) { var curEle = ele.parentNode; while (curEle.nodeName !== 'FORM' && curEle.nodeName !== 'BODY'){ curEle = curEle.parentNode; } return curEle.nodeName === 'FORM' ? curEle : null }; var eles = document.getElementsByClassName('g-recaptcha'); if (eles.length > 0) { var form = closestForm(eles[0]); if (form) { form.submit(); } } }; </script> <input type="submit" data-sitekey="6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj" data-callback="invisibleRecaptchaSubmit" class="g-recaptcha btn btn-primary btn-block" value="Email me a link" value=""/> </form></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/collapse-45805421cf446ca5adf7aaa1935b08a3a8d1d9a6cc5d91a62a2a3a00b20b3e6a.js"], function() { // from javascript_helper.rb $("#login-modal-reset-password-container").on("shown.bs.collapse", function() { $(this).find("input[type=email]").focus(); }); }); </script> </div></div></div><div class="modal-footer"><div class="text-center"><small style="font-size: 12px;">Need an account? <a rel="nofollow" href="https://www.academia.edu/signup">Click here to sign up</a></small></div></div></div></div></div></div><script>// If we are on subdomain or non-bootstrapped page, redirect to login page instead of showing modal (function(){ if (typeof $ === 'undefined') return; var host = window.location.hostname; if ((host === $domain || host === "www."+$domain) && (typeof $().modal === 'function')) { $("#nav_log_in").click(function(e) { // Don't follow the link and open the modal e.preventDefault(); $("#login-modal").on('shown.bs.modal', function() { $(this).find("#login-modal-email-input").focus() }).modal('show'); }); } })()</script> <div class="bootstrap" id="footer"><div class="footer-content clearfix text-center padding-top-7x" style="width:100%;"><ul class="footer-links-secondary footer-links-wide list-inline margin-bottom-1x"><li><a href="https://www.academia.edu/about">About</a></li><li><a href="https://www.academia.edu/press">Press</a></li><li><a rel="nofollow" href="https://medium.com/academia">Blog</a></li><li><a href="https://www.academia.edu/documents">Papers</a></li><li><a href="https://www.academia.edu/topics">Topics</a></li><li><a href="https://www.academia.edu/journals">Academia.edu Journals</a></li><li><a rel="nofollow" href="https://www.academia.edu/hiring"><svg style="width: 13px; height: 13px;" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="briefcase" class="svg-inline--fa fa-briefcase fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M320 336c0 8.84-7.16 16-16 16h-96c-8.84 0-16-7.16-16-16v-48H0v144c0 25.6 22.4 48 48 48h416c25.6 0 48-22.4 48-48V288H320v48zm144-208h-80V80c0-25.6-22.4-48-48-48H176c-25.6 0-48 22.4-48 48v48H48c-25.6 0-48 22.4-48 48v80h512v-80c0-25.6-22.4-48-48-48zm-144 0H192V96h128v32z"></path></svg> <strong>We're Hiring!</strong></a></li><li><a rel="nofollow" href="https://support.academia.edu/"><svg style="width: 12px; height: 12px;" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="question-circle" class="svg-inline--fa fa-question-circle fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M504 256c0 136.997-111.043 248-248 248S8 392.997 8 256C8 119.083 119.043 8 256 8s248 111.083 248 248zM262.655 90c-54.497 0-89.255 22.957-116.549 63.758-3.536 5.286-2.353 12.415 2.715 16.258l34.699 26.31c5.205 3.947 12.621 3.008 16.665-2.122 17.864-22.658 30.113-35.797 57.303-35.797 20.429 0 45.698 13.148 45.698 32.958 0 14.976-12.363 22.667-32.534 33.976C247.128 238.528 216 254.941 216 296v4c0 6.627 5.373 12 12 12h56c6.627 0 12-5.373 12-12v-1.333c0-28.462 83.186-29.647 83.186-106.667 0-58.002-60.165-102-116.531-102zM256 338c-25.365 0-46 20.635-46 46 0 25.364 20.635 46 46 46s46-20.636 46-46c0-25.365-20.635-46-46-46z"></path></svg> <strong>Help Center</strong></a></li></ul><ul class="footer-links-tertiary list-inline margin-bottom-1x"><li class="small">Find new research papers in:</li><li class="small"><a href="https://www.academia.edu/Documents/in/Physics">Physics</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Chemistry">Chemistry</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Biology">Biology</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Health_Sciences">Health Sciences</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Ecology">Ecology</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Earth_Sciences">Earth Sciences</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Cognitive_Science">Cognitive Science</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Computer_Science">Computer Science</a></li></ul></div></div><div class="DesignSystem" id="credit" style="width:100%;"><ul class="u-pl0x footer-links-legal list-inline"><li><a rel="nofollow" href="https://www.academia.edu/terms">Terms</a></li><li><a rel="nofollow" href="https://www.academia.edu/privacy">Privacy</a></li><li><a rel="nofollow" href="https://www.academia.edu/copyright">Copyright</a></li><li>Academia ©2024</li></ul></div><script> //<![CDATA[ window.detect_gmtoffset = true; window.Academia && window.Academia.set_gmtoffset && Academia.set_gmtoffset('/gmtoffset'); //]]> </script> <div id='overlay_background'></div> <div id='bootstrap-modal-container' class='bootstrap'></div> <div id='ds-modal-container' class='bootstrap DesignSystem'></div> <div id='full-screen-modal'></div> </div> </body> </html>