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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" /> <meta name="twitter:image" content="http://a.academia-assets.com/images/twitter-card.jpeg" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/72794607/The_expected_number_of_critical_percolation_clusters_intersecting_a_line_segment" /> <meta property="og:title" content="The expected number of critical percolation clusters intersecting a line segment" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="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" /> <meta property="article:author" content="https://independent.academia.edu/ReneConijn" /> <meta name="description" content="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" /> <title>(PDF) The expected number of critical percolation clusters intersecting a line segment | Rene Conijn - Academia.edu</title> <link rel="canonical" href="https://www.academia.edu/54335806/The_expected_number_of_critical_percolation_clusters_intersecting_a_line_segment" /> <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': "single_work", 'action': "show", 'controller_action': 'single_work#show', '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> var $controller_name = 'single_work'; var $action_name = "show"; var $rails_env = 'production'; var $app_rev = '92477ec68c09d28ae4730a4143c926f074776319'; 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.require = { config: function() { return function() {} } } </script> <script> window.Aedu = window.Aedu || {}; window.Aedu.hit_data = null; window.Aedu.serverRenderTime = new Date(1732781126000); window.Aedu.timeDifference = new Date().getTime() - 1732781126000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","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).","author":[{"@context":"https://schema.org","@type":"Person","name":"Rene Conijn"}],"contributor":[],"dateCreated":"2022-03-02","dateModified":"2022-03-02","datePublished":"2016-03-29","headline":"The expected number of critical percolation clusters intersecting a line segment","inLanguage":"en","keywords":["Mathematics","Statistics"],"locationCreated":null,"publication":null,"publisher":{"@context":"https://schema.org","@type":"Organization","name":null},"image":null,"thumbnailUrl":null,"url":"https://www.academia.edu/72794607/The_expected_number_of_critical_percolation_clusters_intersecting_a_line_segment","sourceOrganization":[{"@context":"https://schema.org","@type":"EducationalOrganization","name":null}]}</script><link rel="stylesheet" media="all" 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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":"2016,3,29"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"The expected number of critical percolation clusters intersecting a line segment","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [102711148]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "full_page_mobile_sutd_modal"; window.loswp.useOptimizedScribd4genScript = false; window.loswp.appleClientId = 'edu.academia.applesignon';</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":81579813,"attachmentType":"pdf"}"><img alt="First page of “The expected number of critical percolation clusters intersecting a line segment”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/81579813/mini_magick20220302-9840-1n54vpp.png?1646231595" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/assets/single_work_splash/adobe.icon-574afd46eb6b03a77a153a647fb47e30546f9215c0ee6a25df597a779717f9ef.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">The expected number of critical percolation clusters intersecting a line segment</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="102711148" href="https://independent.academia.edu/ReneConijn"><img alt="Profile image of Rene Conijn" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Rene Conijn</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2016</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">12 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 72794607; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=72794607"; const getWorkViews = async (workId) => { const response = await fetch(worksViewsPath); if (!response.ok) { throw new Error('Failed to load work views'); } const data = await response.json(); return data.views[workId]; }; // Get the view count for the work - we send this immediately rather than waiting for // the DOM to load, so it can be available as soon as possible (but without holding up // the backend or other resource requests, because it's a bit expensive and not critical). const viewCount = await getWorkViews(workId); const updateViewCount = (viewCount) => { const viewCountNumber = Number(viewCount); if (!viewCountNumber) { throw new Error('Failed to parse view count'); } const commaizedViewCount = viewCountNumber.toLocaleString(); const viewCountBody = document.getElementById('work-metadata-view-count'); if (viewCountBody) { viewCountBody.textContent = `${commaizedViewCount} views`; } else { throw new Error('Failed to find work views element'); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">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).</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":81579813,"attachmentType":"pdf","workUrl":"https://www.academia.edu/72794607/The_expected_number_of_critical_percolation_clusters_intersecting_a_line_segment"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":81579813,"attachmentType":"pdf","workUrl":"https://www.academia.edu/72794607/The_expected_number_of_critical_percolation_clusters_intersecting_a_line_segment"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download 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href="https://independent.academia.edu/BertrandDuplantier">Bertrand Duplantier</a><span>, </span><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="33005068" href="https://independent.academia.edu/MichaelAizenman">Michael Aizenman</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review Letters, 1999</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Path-Crossing Exponents and the External Perimeter in 2D Percolation","attachmentId":44882376,"attachmentType":"pdf","work_url":"https://www.academia.edu/13849166/Path_Crossing_Exponents_and_the_External_Perimeter_in_2D_Percolation","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/13849166/Path_Crossing_Exponents_and_the_External_Perimeter_in_2D_Percolation"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="52412909" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/52412909/Percolation_on_an_isotropically_directed_lattice">Percolation on an isotropically directed lattice</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="61311359" href="https://independent.academia.edu/AureliodeNoronha">Aurelio de Noronha</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Percolation on an isotropically directed lattice","attachmentId":69687548,"attachmentType":"pdf","work_url":"https://www.academia.edu/52412909/Percolation_on_an_isotropically_directed_lattice","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/52412909/Percolation_on_an_isotropically_directed_lattice"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="54335823" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/54335823/Factorization_Formulas_for_2D_Critical_Percolation_Revisited">Factorization Formulas for $2D$ Critical Percolation, Revisited</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="102711148" href="https://independent.academia.edu/ReneConijn">Rene Conijn</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2015</p><p class="ds-related-work--abstract ds2-5-body-sm">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 $s>0$.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Factorization Formulas for $2D$ Critical Percolation, Revisited","attachmentId":70749716,"attachmentType":"pdf","work_url":"https://www.academia.edu/54335823/Factorization_Formulas_for_2D_Critical_Percolation_Revisited","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/54335823/Factorization_Formulas_for_2D_Critical_Percolation_Revisited"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":81579813,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":81579813,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_81579813" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. 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style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/48449807/Site_percolation_on_lattices_with_low_average_coordination_numbers"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="8" data-entity-id="85156478" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/85156478/Universal_features_of_cluster_numbers_in_percolation">Universal features of cluster numbers in percolation</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="160629230" href="https://sfb-trr-62.academia.edu/StephanMertens">Stephan Mertens</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2017</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Universal features of cluster numbers in percolation","attachmentId":89942060,"attachmentType":"pdf","work_url":"https://www.academia.edu/85156478/Universal_features_of_cluster_numbers_in_percolation","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/85156478/Universal_features_of_cluster_numbers_in_percolation"><span class="ds2-5-text-link__content">View 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Hoshen</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Physics A: Mathematical and General, 1979</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Monte Carlo experiments on cluster size distribution in percolation","attachmentId":46873148,"attachmentType":"pdf","work_url":"https://www.academia.edu/26581783/Monte_Carlo_experiments_on_cluster_size_distribution_in_percolation","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/26581783/Monte_Carlo_experiments_on_cluster_size_distribution_in_percolation"><span class="ds2-5-text-link__content">View PDF</span><span 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