CINXE.COM

<!DOCTYPE html> <html > <head> <meta charset="utf-8"> <meta rel="search" type="application/opensearchdescription+xml" href="/open_search.xml" title="Academia.edu"> <meta content="width=device-width, initial-scale=1" name="viewport"> <meta name="google-site-verification" content="bKJMBZA7E43xhDOopFZkssMMkBRjvYERV-NaN4R6mrs"> <meta name="csrf-param" content="authenticity_token" /> <meta name="csrf-token" content="hw2Phon47mLdDuhyznCvIDAi3dK/LbQ2yeuu/1J+lbeGOH+x2gxONVmdWJgWBxNChWx09bL5LPhF62K5A8Tu7A==" /> <meta name="citation_title" content="O ct 2 01 5 Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks" /> <meta name="citation_publication_date" content="2015/01/01" /> <meta name="citation_author" content="Ralph Kenna" /> <meta name="twitter:card" content="summary" /> <meta name="twitter:url" content="https://www.academia.edu/78033772/O_ct_2_01_5_Partition_function_zeros_for_the_Ising_model_on_complete_graphs_and_on_annealed_scale_free_networks" /> <meta name="twitter:title" content="O ct 2 01 5 Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks" /> <meta name="twitter:description" content="We analyze the partition function of the Ising model on graphs of two different types: complete graphs, wherein all nodes are mutually linked and annealed scale-free networks for which the degree distribution decays as P (k) ∼ k. We are interested in" /> <meta name="twitter:image" content="https://0.academia-photos.com/57856370/15395806/16030371/s200_ralph.kenna.jpg" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/78033772/O_ct_2_01_5_Partition_function_zeros_for_the_Ising_model_on_complete_graphs_and_on_annealed_scale_free_networks" /> <meta property="og:title" content="O ct 2 01 5 Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="We analyze the partition function of the Ising model on graphs of two different types: complete graphs, wherein all nodes are mutually linked and annealed scale-free networks for which the degree distribution decays as P (k) ∼ k. We are interested in" /> <meta property="article:author" content="https://independent.academia.edu/KennaRalph" /> <meta name="description" content="We analyze the partition function of the Ising model on graphs of two different types: complete graphs, wherein all nodes are mutually linked and annealed scale-free networks for which the degree distribution decays as P (k) ∼ k. We are interested in" /> <title>(PDF) O ct 2 01 5 Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks | Ralph Kenna - Academia.edu</title> <link rel="canonical" href="https://www.academia.edu/78033772/O_ct_2_01_5_Partition_function_zeros_for_the_Ising_model_on_complete_graphs_and_on_annealed_scale_free_networks" /> <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 = '49879c2402910372f4abc62630a427bbe033d190'; 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(1732460318000); window.Aedu.timeDifference = new Date().getTime() - 1732460318000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"We analyze the partition function of the Ising model on graphs of two different types: complete graphs, wherein all nodes are mutually linked and annealed scale-free networks for which the degree distribution decays as P (k) ∼ k. We are interested in zeros of the partition function in the cases of complex temperature or complex external field (Fisher and Lee-Yang zeros respectively). For the model on an annealed scale-free network, we find an integral representation for the partition function which, in the case λ \u0026amp;amp;gt; 5, reproduces the zeros for the Ising model on a complete graph. For 3 \u0026amp;amp;lt; λ \u0026amp;amp;lt; 5 we derive the λ-dependent angle at which the Fisher zeros impact onto the real temperature axis. This, in turn, gives access to the λ-dependent universal values of the critical exponents and critical amplitudes ratios. Our analysis of the Lee-Yang zeros reveals a difference in their behaviour for the Ising model on a complete graph and on an annealed scale-free network when 3 \u0026amp;amp;lt; λ \u0026amp;amp;l...","author":[{"@context":"https://schema.org","@type":"Person","name":"Ralph Kenna"}],"contributor":[],"dateCreated":"2022-04-30","dateModified":"2022-04-30","datePublished":"2015-01-01","headline":"O ct 2 01 5 Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks","inLanguage":"en","keywords":[],"locationCreated":null,"publication":null,"publisher":{"@context":"https://schema.org","@type":"Organization","name":null},"image":null,"thumbnailUrl":null,"url":"https://www.academia.edu/78033772/O_ct_2_01_5_Partition_function_zeros_for_the_Ising_model_on_complete_graphs_and_on_annealed_scale_free_networks","sourceOrganization":[{"@context":"https://schema.org","@type":"EducationalOrganization","name":null}]}</script><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/single_work_page/loswp-352e32ba4e89304dc0b4fa5b3952eef2198174c54cdb79066bc62e91c68a1a91.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/body-8d679e925718b5e8e4b18e9a4fab37f7eaa99e43386459376559080ac8f2856a.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/button-3cea6e0ad4715ed965c49bfb15dedfc632787b32ff6d8c3a474182b231146ab7.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/text_button-73590134e40cdb49f9abdc8e796cc00dc362693f3f0f6137d6cf9bb78c318ce7.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" /> </head> <body> <div id='react-modal'></div> <div class="js-upgrade-ie-banner" style="display: none; text-align: center; padding: 8px 0; background-color: #ebe480;"><p style="color: #000; font-size: 12px; margin: 0 0 4px;">Academia.edu no longer supports Internet Explorer.</p><p style="color: #000; font-size: 12px; margin: 0;">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.querySelector('.js-upgrade-ie-banner').style.display = 'block'; }</script> <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: "4d54fe1a9575714092adb86ad2a6854df04c39ef7e905be2a351576d99288e38", });</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="VsSF//7mDATiIBQogT3VJQ2BD8YvaAowMg7O+VIwOwVX8XXIrRKsU2azpMJZSmlHuM+m4SK8kv6+DgK/A4pAXg==" 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://www.academia.edu/78033772/O_ct_2_01_5_Partition_function_zeros_for_the_Ising_model_on_complete_graphs_and_on_annealed_scale_free_networks" 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="URt6t19zjnPVsfYKObOysJYX6hbPIrz2AkZ8hLNdKgNQLoqADIcuJFEiRuDhxA7SI1lDMcL2JDiORrDC4udRWA==" 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><input class="btn btn-primary btn-block g-recaptcha js-password-reset-submit" data-sitekey="6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj" type="submit" value="Email me a link" /></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 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> <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> <div class="header--container" id="main-header-container"><div class="header--inner-container header--inner-container-ds2"><div class="header-ds2--left-wrapper"><div class="header-ds2--left-wrapper-inner"><a data-main-header-link-target="logo_home" href="https://www.academia.edu/"><img class="hide-on-desktop-redesign" style="height: 24px; width: 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="hide-on-mobile-redesign" style="height: 24px;" alt="Academia.edu" src="//a.academia-assets.com/images/academia-logo-redesign-2015.svg" /></a><div class="header--search-container header--search-container-ds2"><form class="js-SiteSearch-form select2-no-default-pills" action="https://www.academia.edu/search" accept-charset="UTF-8" method="get"><input name="utf8" type="hidden" value="✓" autocomplete="off" /><svg style="width: 14px; height: 14px;" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="search" class="header--search-icon svg-inline--fa fa-search fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M505 442.7L405.3 343c-4.5-4.5-10.6-7-17-7H372c27.6-35.3 44-79.7 44-128C416 93.1 322.9 0 208 0S0 93.1 0 208s93.1 208 208 208c48.3 0 92.7-16.4 128-44v16.3c0 6.4 2.5 12.5 7 17l99.7 99.7c9.4 9.4 24.6 9.4 33.9 0l28.3-28.3c9.4-9.4 9.4-24.6.1-34zM208 336c-70.7 0-128-57.2-128-128 0-70.7 57.2-128 128-128 70.7 0 128 57.2 128 128 0 70.7-57.2 128-128 128z"></path></svg><input class="header--search-input header--search-input-ds2 js-SiteSearch-form-input" data-main-header-click-target="search_input" name="q" placeholder="Search" type="text" /></form></div></div></div><nav class="header--nav-buttons header--nav-buttons-ds2 js-main-nav"><a class="ds2-5-button ds2-5-button--secondary js-header-login-url header-button-ds2 header-login-ds2 hide-on-mobile-redesign" href="https://www.academia.edu/login" rel="nofollow">Log In</a><a class="ds2-5-button ds2-5-button--secondary header-button-ds2 hide-on-mobile-redesign" href="https://www.academia.edu/signup" rel="nofollow">Sign Up</a><button class="header--hamburger-button header--hamburger-button-ds2 hide-on-desktop-redesign js-header-hamburger-button"><div class="icon-bar"></div><div class="icon-bar" style="margin-top: 4px;"></div><div class="icon-bar" style="margin-top: 4px;"></div></button></nav></div><ul class="header--dropdown-container js-header-dropdown"><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/login" rel="nofollow">Log In</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/signup" rel="nofollow">Sign Up</a></li><li class="header--dropdown-row js-header-dropdown-expand-button"><button class="header--dropdown-button">more<svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="caret-down" class="header--dropdown-button-icon svg-inline--fa fa-caret-down fa-w-10" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path fill="currentColor" d="M31.3 192h257.3c17.8 0 26.7 21.5 14.1 34.1L174.1 354.8c-7.8 7.8-20.5 7.8-28.3 0L17.2 226.1C4.6 213.5 13.5 192 31.3 192z"></path></svg></button></li><li><ul class="header--expanded-dropdown-container"><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/about">About</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/press">Press</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://medium.com/@academia">Blog</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/documents">Papers</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/terms">Terms</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/privacy">Privacy</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/copyright">Copyright</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/hiring"><svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="briefcase" class="header--dropdown-row-icon 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>We're Hiring!</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://support.academia.edu/"><svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="question-circle" class="header--dropdown-row-icon 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>Help Center</a></li><li class="header--dropdown-row js-header-dropdown-collapse-button"><button class="header--dropdown-button">less<svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="caret-up" class="header--dropdown-button-icon svg-inline--fa fa-caret-up fa-w-10" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path fill="currentColor" d="M288.662 352H31.338c-17.818 0-26.741-21.543-14.142-34.142l128.662-128.662c7.81-7.81 20.474-7.81 28.284 0l128.662 128.662c12.6 12.599 3.676 34.142-14.142 34.142z"></path></svg></button></li></ul></li></ul></div> <script src="//a.academia-assets.com/assets/webpack_bundles/fast_loswp-bundle-bf3d831cde46cd0e142f29f81a3fc4ce5ab45a404c10c12a480e83de68aff851.js" defer="defer"></script><script>window.loswp = {}; window.loswp.author = 57856370; window.loswp.bulkDownloadFilterCounts = {}; window.loswp.hasDownloadableAttachment = true; window.loswp.hasViewableAttachments = true; // TODO: just use routes for this window.loswp.loginUrl = "https://www.academia.edu/login?post_login_redirect_url=https%3A%2F%2Fwww.academia.edu%2F78033772%2FO_ct_2_01_5_Partition_function_zeros_for_the_Ising_model_on_complete_graphs_and_on_annealed_scale_free_networks%3Fauto%3Ddownload"; window.loswp.translateUrl = "https://www.academia.edu/login?post_login_redirect_url=https%3A%2F%2Fwww.academia.edu%2F78033772%2FO_ct_2_01_5_Partition_function_zeros_for_the_Ising_model_on_complete_graphs_and_on_annealed_scale_free_networks%3Fshow_translation%3Dtrue"; window.loswp.previewableAttachments = [{"id":85220221,"identifier":"Attachment_85220221","shouldShowBulkDownload":false}]; window.loswp.shouldDetectTimezone = true; window.loswp.shouldShowBulkDownload = true; window.loswp.showSignupCaptcha = false window.loswp.willEdgeCache = false; window.loswp.work = {"work":{"id":78033772,"created_at":"2022-04-30T04:19:26.364-07:00","from_world_paper_id":203786081,"updated_at":"2022-04-30T06:53:08.619-07:00","_data":{"abstract":"We analyze the partition function of the Ising model on graphs of two different types: complete graphs, wherein all nodes are mutually linked and annealed scale-free networks for which the degree distribution decays as P (k) ∼ k. We are interested in zeros of the partition function in the cases of complex temperature or complex external field (Fisher and Lee-Yang zeros respectively). For the model on an annealed scale-free network, we find an integral representation for the partition function which, in the case λ \u0026gt; 5, reproduces the zeros for the Ising model on a complete graph. For 3 \u0026lt; λ \u0026lt; 5 we derive the λ-dependent angle at which the Fisher zeros impact onto the real temperature axis. This, in turn, gives access to the λ-dependent universal values of the critical exponents and critical amplitudes ratios. Our analysis of the Lee-Yang zeros reveals a difference in their behaviour for the Ising model on a complete graph and on an annealed scale-free network when 3 \u0026lt; λ \u0026l...","publication_date":"2015,,"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"O ct 2 01 5 Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [57856370]; 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":85220221,"attachmentType":"pdf"}"><img alt="First page of “O ct 2 01 5 Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/85220221/mini_magick20220430-2282-1aunh52.png?1651317634" /><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">O ct 2 01 5 Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks</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="57856370" href="https://independent.academia.edu/KennaRalph"><img alt="Profile image of Ralph Kenna" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/57856370/15395806/16030371/s65_ralph.kenna.jpg" />Ralph Kenna</a></div><p class="ds-work-card--detail ds2-5-body-sm">2015</p><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">We analyze the partition function of the Ising model on graphs of two different types: complete graphs, wherein all nodes are mutually linked and annealed scale-free networks for which the degree distribution decays as P (k) ∼ k. We are interested in zeros of the partition function in the cases of complex temperature or complex external field (Fisher and Lee-Yang zeros respectively). For the model on an annealed scale-free network, we find an integral representation for the partition function which, in the case λ &gt; 5, reproduces the zeros for the Ising model on a complete graph. For 3 &lt; λ &lt; 5 we derive the λ-dependent angle at which the Fisher zeros impact onto the real temperature axis. This, in turn, gives access to the λ-dependent universal values of the critical exponents and critical amplitudes ratios. Our analysis of the Lee-Yang zeros reveals a difference in their behaviour for the Ising model on a complete graph and on an annealed scale-free network when 3 &lt; λ &l...</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":85220221,"attachmentType":"pdf","workUrl":"https://www.academia.edu/78033772/O_ct_2_01_5_Partition_function_zeros_for_the_Ising_model_on_complete_graphs_and_on_annealed_scale_free_networks"}">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":85220221,"attachmentType":"pdf","workUrl":"https://www.academia.edu/78033772/O_ct_2_01_5_Partition_function_zeros_for_the_Ising_model_on_complete_graphs_and_on_annealed_scale_free_networks"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div></div><div data-auto_select="false" data-client_id="331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b" data-doc_id="85220221" data-landing_url="https://www.academia.edu/78033772/O_ct_2_01_5_Partition_function_zeros_for_the_Ising_model_on_complete_graphs_and_on_annealed_scale_free_networks" data-login_uri="https://www.academia.edu/registrations/google_one_tap" data-moment_callback="onGoogleOneTapEvent" id="g_id_onload"></div><div class="ds-top-related-works--grid-container"><div class="ds-related-content--container ds-top-related-works--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="0" data-entity-id="78033765" 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/78033765/Partition_function_zeros_for_the_Ising_model_on_complete_graphs_and_on_annealed_scale_free_networks">Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks</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="57856370" href="https://independent.academia.edu/KennaRalph">Ralph Kenna</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Physics A: Mathematical and Theoretical, 2016</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":"Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks","attachmentId":85220362,"attachmentType":"pdf","work_url":"https://www.academia.edu/78033765/Partition_function_zeros_for_the_Ising_model_on_complete_graphs_and_on_annealed_scale_free_networks","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/78033765/Partition_function_zeros_for_the_Ising_model_on_complete_graphs_and_on_annealed_scale_free_networks"><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="1" data-entity-id="78033770" 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/78033770/Generalized_Ising_Model_on_a_Scale_Free_Network_An_Interplay_of_Power_Laws">Generalized Ising Model on a Scale-Free Network: An Interplay of Power Laws</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="57856370" href="https://independent.academia.edu/KennaRalph">Ralph Kenna</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Entropy, 2021</p><p class="ds-related-work--abstract ds2-5-body-sm">We consider a recently introduced generalization of the Ising model in which individual spin strength can vary. The model is intended for analysis of ordering in systems comprising agents which, although matching in their binarity (i.e., maintaining the iconic Ising features of ‘+’ or ‘−’, ‘up’ or ‘down’, ‘yes’ or ‘no’), differ in their strength. To investigate the interplay between variable properties of nodes and interactions between them, we study the model on a complex network where both the spin strength and degree distributions are governed by power laws. We show that in the annealed network approximation, thermodynamic functions of the model are self-averaging and we obtain an exact solution for the partition function. This allows us derive the leading temperature and field dependencies of thermodynamic functions, their critical behavior, and logarithmic corrections at the interface of different phases. We find the delicate interplay of the two power laws leads to new univers...</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":"Generalized Ising Model on a Scale-Free Network: An Interplay of Power Laws","attachmentId":85220215,"attachmentType":"pdf","work_url":"https://www.academia.edu/78033770/Generalized_Ising_Model_on_a_Scale_Free_Network_An_Interplay_of_Power_Laws","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/78033770/Generalized_Ising_Model_on_a_Scale_Free_Network_An_Interplay_of_Power_Laws"><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="2" data-entity-id="11049486" 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/11049486/Yang_Lee_zeros_of_the_Ising_model_on_random_graphs_of_non_planar_topology">Yang–Lee zeros of the Ising model on random graphs of non planar topology</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="26727117" href="https://independent.academia.edu/NelsonAlves6">Nelson Alves</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Nuclear Physics B, 2000</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":"Yang–Lee zeros of the Ising model on random graphs of non planar topology","attachmentId":46945193,"attachmentType":"pdf","work_url":"https://www.academia.edu/11049486/Yang_Lee_zeros_of_the_Ising_model_on_random_graphs_of_non_planar_topology","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/11049486/Yang_Lee_zeros_of_the_Ising_model_on_random_graphs_of_non_planar_topology"><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="3" data-entity-id="15962277" 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/15962277/Violation_of_Lee_Yang_circle_theorem_for_Ising_phase_transitions_on_complex_networks">Violation of Lee-Yang circle theorem for Ising phase transitions on complex networks</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="35114594" href="https://independent.academia.edu/MKrasnytska">M. Krasnytska</a></div><p class="ds-related-work--abstract ds2-5-body-sm">The Ising model on annealed complex networks with degree distribution decaying algebraically as $p(K)\sim K^{-\lambda}$ has a second-order phase transition at finite temperature if $\lambda&gt; 3$. In the absence of space dimensionality, $\lambda$ controls the transition strength; mean-field theory applies for $\lambda &gt;5$ but critical exponents are $\lambda$-dependent if $\lambda &lt; 5$. Here we show that, as for regular lattices, the celebrated Lee-Yang circle theorem is obeyed for the former case. However, unlike on regular lattices where it is independent of dimensionality, the circle theorem fails on complex networks when $\lambda &lt; 5$. We discuss the importance of this result for both theory and experiments on phase transitions and critical phenomena. We also investigate the finite-size scaling of Lee-Yang zeros in both regimes as well as the multiplicative logarithmic corrections which occur at $\lambda=5$.</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":"Violation of Lee-Yang circle theorem for Ising phase transitions on complex networks","attachmentId":41208042,"attachmentType":"pdf","work_url":"https://www.academia.edu/15962277/Violation_of_Lee_Yang_circle_theorem_for_Ising_phase_transitions_on_complex_networks","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/15962277/Violation_of_Lee_Yang_circle_theorem_for_Ising_phase_transitions_on_complex_networks"><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="4" data-entity-id="16409445" 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/16409445/Zeros_of_the_partition_function_and_pseudospinodals_in_long_range_Ising_models">Zeros of the partition function and pseudospinodals in long-range Ising models</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="35553607" href="https://bgi.academia.edu/NataliGulbahce">Natali Gulbahce</a><span>, </span><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="35641573" href="https://independent.academia.edu/WilliamKlein2">William Klein</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2004</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":"Zeros of the partition function and pseudospinodals in long-range Ising models","attachmentId":42495004,"attachmentType":"pdf","work_url":"https://www.academia.edu/16409445/Zeros_of_the_partition_function_and_pseudospinodals_in_long_range_Ising_models","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/16409445/Zeros_of_the_partition_function_and_pseudospinodals_in_long_range_Ising_models"><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="5" data-entity-id="110662538" 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/110662538/Ising_models_on_Feynman_graphs">Ising models on Feynman graphs</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="39738157" href="https://unimi.academia.edu/LucaGuidoMolinari">Luca Guido Molinari</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Nuclear Physics B, 1988</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":"Ising models on Feynman graphs","attachmentId":108410376,"attachmentType":"pdf","work_url":"https://www.academia.edu/110662538/Ising_models_on_Feynman_graphs","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/110662538/Ising_models_on_Feynman_graphs"><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="6" data-entity-id="56610024" 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/56610024/Density_of_the_Fisher_Zeroes_for_the_Ising_Model">Density of the Fisher Zeroes for the Ising Model</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="509143" href="https://neu.academia.edu/WentaoLu">Wentao Lu</a></div><p class="ds-related-work--metadata ds2-5-body-xs">J Statist Phys, 2001</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":"Density of the Fisher Zeroes for the Ising Model","attachmentId":71912444,"attachmentType":"pdf","work_url":"https://www.academia.edu/56610024/Density_of_the_Fisher_Zeroes_for_the_Ising_Model","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/56610024/Density_of_the_Fisher_Zeroes_for_the_Ising_Model"><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="7" data-entity-id="88895921" 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/88895921/Critical_behavior_of_the_Ising_model_in_annealed_scale_free_networks">Critical behavior of the Ising model in annealed scale-free networks</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="65884543" href="https://independent.academia.edu/ParkHyunggyu">Hyunggyu Park</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2009</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":"Critical behavior of the Ising model in annealed scale-free networks","attachmentId":92792933,"attachmentType":"pdf","work_url":"https://www.academia.edu/88895921/Critical_behavior_of_the_Ising_model_in_annealed_scale_free_networks","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/88895921/Critical_behavior_of_the_Ising_model_in_annealed_scale_free_networks"><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="81159086" 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/81159086/Non_equilibrium_critical_properties_of_the_Ising_model_on_product_graphs">Non-equilibrium critical properties of the Ising model on product graphs</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="34467673" href="https://independent.academia.edu/FCorberi">F. Corberi</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Statistical Mechanics: Theory and Experiment, 2010</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":"Non-equilibrium critical properties of the Ising model on product graphs","attachmentId":87301288,"attachmentType":"pdf","work_url":"https://www.academia.edu/81159086/Non_equilibrium_critical_properties_of_the_Ising_model_on_product_graphs","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/81159086/Non_equilibrium_critical_properties_of_the_Ising_model_on_product_graphs"><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="8898543" 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/8898543/Phase_transitions_in_an_Ising_model_on_a_Euclidean_network">Phase transitions in an Ising model on a Euclidean network</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="19572780" href="https://caluniv.academia.edu/ParongamaSen">Parongama Sen</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2006</p><p class="ds-related-work--abstract ds2-5-body-sm">A one dimensional network on which there are long range bonds at lattice distances $l>1$ with the probability $P(l) \propto l^{-\delta}$ has been taken under consideration. We investigate the critical behavior of the Ising model on such a network where spins interact with these extra neighbours apart from their nearest neighbours for $0 \leq \delta < 2$. It is observed that there is a finite temperature phase transition in the entire range. For $0 \leq \delta < 1$, finite size scaling behaviour of various quantities are consistent with mean field exponents while for $1\leq \delta\leq 2$, the exponents depend on $\delta$. The results are discussed in the context of earlier observations on the topology of the underlying network.</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":"Phase transitions in an Ising model on a Euclidean network","attachmentId":35227887,"attachmentType":"pdf","work_url":"https://www.academia.edu/8898543/Phase_transitions_in_an_Ising_model_on_a_Euclidean_network","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/8898543/Phase_transitions_in_an_Ising_model_on_a_Euclidean_network"><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":85220221,"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":85220221,"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_85220221" 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. You can download the paper by clicking the button above.</p></div></div></div></div><div class="ds-sidebar--container js-work-sidebar"><div class="ds-related-content--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="0" data-entity-id="20594517" 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/20594517/Non_equilibrium_phase_transitions_of_the_Ising_model_on_networks">Non equilibrium phase transitions of the Ising model on networks</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="19572780" href="https://caluniv.academia.edu/ParongamaSen">Parongama Sen</a></div><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":"Non equilibrium phase transitions of the Ising model on networks","attachmentId":41455793,"attachmentType":"pdf","work_url":"https://www.academia.edu/20594517/Non_equilibrium_phase_transitions_of_the_Ising_model_on_networks","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/20594517/Non_equilibrium_phase_transitions_of_the_Ising_model_on_networks"><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="1" data-entity-id="14031828" 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/14031828/Ising_Critical_Exponents_on_Random_Trees_and_Graphs">Ising Critical Exponents on Random Trees and Graphs</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="33059620" href="https://unimore.academia.edu/CristianGiardina">Cristian Giardina</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Communications in Mathematical Physics, 2014</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":"Ising Critical Exponents on Random Trees and Graphs","attachmentId":44684940,"attachmentType":"pdf","work_url":"https://www.academia.edu/14031828/Ising_Critical_Exponents_on_Random_Trees_and_Graphs","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/14031828/Ising_Critical_Exponents_on_Random_Trees_and_Graphs"><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="2" data-entity-id="22152665" 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/22152665/Complex_zeros_of_the_2d_Ising_model_on_dynamical_random_lattices">Complex zeros of the 2d Ising model on dynamical random lattices</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="4098" href="https://ntua.academia.edu/KonstantinosAnagnostopoulos">Konstantinos Anagnostopoulos</a><span>, </span><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32981670" href="https://independent.academia.edu/JAmbj%C3%B8rn">J. Ambjørn</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Nuclear Physics B - Proceedings Supplements, 1998</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":"Complex zeros of the 2d Ising model on dynamical random lattices","attachmentId":42815052,"attachmentType":"pdf","work_url":"https://www.academia.edu/22152665/Complex_zeros_of_the_2d_Ising_model_on_dynamical_random_lattices","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/22152665/Complex_zeros_of_the_2d_Ising_model_on_dynamical_random_lattices"><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="3" data-entity-id="7910200" 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/7910200/Partition_function_zeros_of_the_antiferromagnetic_Ising_model_on_triangular_lattice_in_the_complex_temperature_plane_for_nonzero_magnetic_field">Partition function zeros of the antiferromagnetic Ising model on triangular lattice in the complex temperature plane for nonzero magnetic field</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="14826962" href="https://gist.academia.edu/ChiOkHwang">Chi-Ok Hwang</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Nuclear Physics B, 2008</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":"Partition function zeros of the antiferromagnetic Ising model on triangular lattice in the complex temperature plane for nonzero magnetic field","attachmentId":48293553,"attachmentType":"pdf","work_url":"https://www.academia.edu/7910200/Partition_function_zeros_of_the_antiferromagnetic_Ising_model_on_triangular_lattice_in_the_complex_temperature_plane_for_nonzero_magnetic_field","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/7910200/Partition_function_zeros_of_the_antiferromagnetic_Ising_model_on_triangular_lattice_in_the_complex_temperature_plane_for_nonzero_magnetic_field"><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="4" data-entity-id="114095757" 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/114095757/Finite_size_scaling_functions_of_the_phase_transition_in_the_ferromagnetic_Ising_model_on_random_regular_graphs">Finite size scaling functions of the phase transition in the ferromagnetic Ising model on random regular graphs</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="231144340" href="https://independent.academia.edu/DeepakDhar4">Deepak Dhar</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Statistical Mechanics: Theory and Experiment</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":"Finite size scaling functions of the phase transition in the ferromagnetic Ising model on random regular graphs","attachmentId":110884004,"attachmentType":"pdf","work_url":"https://www.academia.edu/114095757/Finite_size_scaling_functions_of_the_phase_transition_in_the_ferromagnetic_Ising_model_on_random_regular_graphs","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/114095757/Finite_size_scaling_functions_of_the_phase_transition_in_the_ferromagnetic_Ising_model_on_random_regular_graphs"><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="5" data-entity-id="68601940" 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/68601940/Asymptotics_of_the_partition_function_of_Ising_model_on_inhomogeneous_random_graphs">Asymptotics of the partition function of Ising model on inhomogeneous random graphs</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="203442452" href="https://independent.academia.edu/KwabenaDokuAmponsah">Kwabena Doku-Amponsah</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2012</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":"Asymptotics of the partition function of Ising model on inhomogeneous random graphs","attachmentId":79026024,"attachmentType":"pdf","work_url":"https://www.academia.edu/68601940/Asymptotics_of_the_partition_function_of_Ising_model_on_inhomogeneous_random_graphs","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/68601940/Asymptotics_of_the_partition_function_of_Ising_model_on_inhomogeneous_random_graphs"><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="6" data-entity-id="32867576" 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/32867576/Fat_and_thin_Fisher_zeroes">Fat and thin Fisher zeroes</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="23419258" href="https://hw.academia.edu/DJohnston">Des Johnston</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Nuclear Physics B - Proceedings Supplements, 2002</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":"Fat and thin Fisher zeroes","attachmentId":53013011,"attachmentType":"pdf","work_url":"https://www.academia.edu/32867576/Fat_and_thin_Fisher_zeroes","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/32867576/Fat_and_thin_Fisher_zeroes"><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="7" data-entity-id="11833673" 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/11833673/A_new_look_at_the_2D_Ising_model_from_exact_partition_function_zeros_for_large_lattice_sizes">A new look at the 2D Ising model from exact partition function zeros for large lattice sizes</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="29198387" href="https://usp-br.academia.edu/NelsonAlves">Nelson Alves</a></div><p class="ds-related-work--metadata ds2-5-body-xs">1997</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":"A new look at the 2D Ising model from exact partition function zeros for large lattice sizes","attachmentId":46496575,"attachmentType":"pdf","work_url":"https://www.academia.edu/11833673/A_new_look_at_the_2D_Ising_model_from_exact_partition_function_zeros_for_large_lattice_sizes","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/11833673/A_new_look_at_the_2D_Ising_model_from_exact_partition_function_zeros_for_large_lattice_sizes"><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="55613252" 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/55613252/Ising_Models_on_Static_Inhomogeneous_Random_graphs">Ising Models on Static Inhomogeneous Random graphs</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="203442452" href="https://independent.academia.edu/KwabenaDokuAmponsah">Kwabena Doku-Amponsah</a></div><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":"Ising Models on Static Inhomogeneous Random graphs","attachmentId":71403501,"attachmentType":"pdf","work_url":"https://www.academia.edu/55613252/Ising_Models_on_Static_Inhomogeneous_Random_graphs","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/55613252/Ising_Models_on_Static_Inhomogeneous_Random_graphs"><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="9" data-entity-id="32867551" 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/32867551/Thin_Fisher_zeros">Thin Fisher zeros</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="23419258" href="https://hw.academia.edu/DJohnston">Des Johnston</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Physics A: Mathematical and General, 2001</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":"Thin Fisher zeros","attachmentId":53012987,"attachmentType":"pdf","work_url":"https://www.academia.edu/32867551/Thin_Fisher_zeros","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/32867551/Thin_Fisher_zeros"><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="10" data-entity-id="78838514" 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/78838514/Finite_size_effects_for_the_Ising_model_on_random_graphs_with_varying_dilution">Finite size effects for the Ising model on random graphs with varying dilution</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="8338290" href="https://unifi.academia.edu/FrancoBagnoli">Franco Bagnoli</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physica A: Statistical Mechanics and its Applications, 2009</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":"Finite size effects for the Ising model on random graphs with varying dilution","attachmentId":85741272,"attachmentType":"pdf","work_url":"https://www.academia.edu/78838514/Finite_size_effects_for_the_Ising_model_on_random_graphs_with_varying_dilution","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/78838514/Finite_size_effects_for_the_Ising_model_on_random_graphs_with_varying_dilution"><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="11" data-entity-id="122452553" 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/122452553/Yang_Lee_and_Fisher_zeros_of_multisite_interaction_Ising_models_on_the_Cayley_type_lattices">Yang–Lee and Fisher zeros of multisite interaction Ising models on the Cayley-type lattices</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="38166739" href="https://independent.academia.edu/NAnanikian">N. Ananikian</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physics Letters A, 2000</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":"Yang–Lee and Fisher zeros of multisite interaction Ising models on the Cayley-type lattices","attachmentId":117112979,"attachmentType":"pdf","work_url":"https://www.academia.edu/122452553/Yang_Lee_and_Fisher_zeros_of_multisite_interaction_Ising_models_on_the_Cayley_type_lattices","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/122452553/Yang_Lee_and_Fisher_zeros_of_multisite_interaction_Ising_models_on_the_Cayley_type_lattices"><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="12" data-entity-id="14031841" 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/14031841/Ising_Models_on_Power_Law_Random_Graphs">Ising Models on Power-Law Random Graphs</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="33059620" href="https://unimore.academia.edu/CristianGiardina">Cristian Giardina</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Statistical Physics, 2010</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":"Ising Models on Power-Law Random Graphs","attachmentId":44684936,"attachmentType":"pdf","work_url":"https://www.academia.edu/14031841/Ising_Models_on_Power_Law_Random_Graphs","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/14031841/Ising_Models_on_Power_Law_Random_Graphs"><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="13" data-entity-id="8898566" 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/8898566/Zero_temperature_dynamics_of_Ising_model_on_a_densely_connected_small_world_network">Zero temperature dynamics of Ising model on a densely connected small world network</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="19572780" href="https://caluniv.academia.edu/ParongamaSen">Parongama Sen</a></div><p class="ds-related-work--metadata ds2-5-body-xs">European Physical Journal B, 2005</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":"Zero temperature dynamics of Ising model on a densely connected small world network","attachmentId":47961282,"attachmentType":"pdf","work_url":"https://www.academia.edu/8898566/Zero_temperature_dynamics_of_Ising_model_on_a_densely_connected_small_world_network","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/8898566/Zero_temperature_dynamics_of_Ising_model_on_a_densely_connected_small_world_network"><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="14" data-entity-id="40929425" 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/40929425/Fisher_zeros_in_the_Kallen_Lehmann_approach_to_3D_Ising_model">Fisher zeros in the Kallen–Lehmann approach to 3D Ising model</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="135164713" href="https://nyu.academia.edu/GastonGiribet">Gaston Giribet</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physics Letters B, 2009</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":"Fisher zeros in the Kallen–Lehmann approach to 3D Ising model","attachmentId":61218553,"attachmentType":"pdf","work_url":"https://www.academia.edu/40929425/Fisher_zeros_in_the_Kallen_Lehmann_approach_to_3D_Ising_model","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/40929425/Fisher_zeros_in_the_Kallen_Lehmann_approach_to_3D_Ising_model"><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="15" data-entity-id="122741478" 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/122741478/Exploring_Lee_Yang_and_Fisher_Zeros_in_the_2D_Ising_Model_through_Multi_Point_Pad_e_Approximants">Exploring Lee-Yang and Fisher Zeros in the 2D Ising Model through Multi-Point Pad\'e Approximants</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="271644068" href="https://independent.academia.edu/MassimoCipressi">Massimo Cipressi</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2023</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":"Exploring Lee-Yang and Fisher Zeros in the 2D Ising Model through Multi-Point Pad\\'e Approximants","attachmentId":117344467,"attachmentType":"pdf","work_url":"https://www.academia.edu/122741478/Exploring_Lee_Yang_and_Fisher_Zeros_in_the_2D_Ising_Model_through_Multi_Point_Pad_e_Approximants","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/122741478/Exploring_Lee_Yang_and_Fisher_Zeros_in_the_2D_Ising_Model_through_Multi_Point_Pad_e_Approximants"><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="16" data-entity-id="10331104" 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/10331104/Intrinsic_degree_correlations_in_the_static_model_of_scale_free_networks">Intrinsic degree-correlations in the static model of scale-free networks</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="25329020" href="https://independent.academia.edu/GohKS">KS Goh</a></div><p class="ds-related-work--metadata ds2-5-body-xs">European Physical Journal B, 2006</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":"Intrinsic degree-correlations in the static model of scale-free networks","attachmentId":47434781,"attachmentType":"pdf","work_url":"https://www.academia.edu/10331104/Intrinsic_degree_correlations_in_the_static_model_of_scale_free_networks","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/10331104/Intrinsic_degree_correlations_in_the_static_model_of_scale_free_networks"><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="17" data-entity-id="94168047" 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/94168047/The_bivariate_Ising_polynomial_of_a_graph">The bivariate Ising polynomial of a graph</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="251735137" href="https://independent.academia.edu/DanielAndr%C3%A9n">Daniel Andrén</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Discrete Applied Mathematics, 2009</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":"The bivariate Ising polynomial of a graph","attachmentId":96701100,"attachmentType":"pdf","work_url":"https://www.academia.edu/94168047/The_bivariate_Ising_polynomial_of_a_graph","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/94168047/The_bivariate_Ising_polynomial_of_a_graph"><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="18" data-entity-id="32867616" 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/32867616/Information_geometry_of_the_ising_model_on_planar_random_graphs">Information geometry of the ising model on planar random graphs</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="23419258" href="https://hw.academia.edu/DJohnston">Des Johnston</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 2002</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":"Information geometry of the ising model on planar random graphs","attachmentId":53013049,"attachmentType":"pdf","work_url":"https://www.academia.edu/32867616/Information_geometry_of_the_ising_model_on_planar_random_graphs","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/32867616/Information_geometry_of_the_ising_model_on_planar_random_graphs"><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="19" data-entity-id="4819112" 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/4819112/Complex_Networks_Renormalization_Flows_and_Fixed_Points">Complex Networks Renormalization: Flows and Fixed Points</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="6230751" href="https://independent.academia.edu/FilippoRadicchi">Filippo Radicchi</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review Letters, 2008</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":"Complex Networks Renormalization: Flows and Fixed Points","attachmentId":49616985,"attachmentType":"pdf","work_url":"https://www.academia.edu/4819112/Complex_Networks_Renormalization_Flows_and_Fixed_Points","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/4819112/Complex_Networks_Renormalization_Flows_and_Fixed_Points"><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="20" data-entity-id="50726397" 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/50726397/Singularities_of_the_Partition_Function_for_the_Ising_Model_Coupled_to_2D_Quantum_Gravity">Singularities of the Partition Function for the Ising Model Coupled to 2D Quantum Gravity</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="58304863" href="https://independent.academia.edu/JanAmbjorn">Jan Ambjorn</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Modern Physics Letters A, 1997</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":"Singularities of the Partition Function for the Ising Model Coupled to 2D Quantum Gravity","attachmentId":68598563,"attachmentType":"pdf","work_url":"https://www.academia.edu/50726397/Singularities_of_the_Partition_Function_for_the_Ising_Model_Coupled_to_2D_Quantum_Gravity","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/50726397/Singularities_of_the_Partition_Function_for_the_Ising_Model_Coupled_to_2D_Quantum_Gravity"><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="21" data-entity-id="69177743" 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/69177743/The_high_temperature_Ising_model_on_the_triangular_lattice_is_a_critical_percolation_model">The high temperature Ising model on the triangular lattice is a critical percolation model</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="52848710" href="https://independent.academia.edu/FedericoCamia">Federico Camia</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Eprint Arxiv 0806 3020, 2008</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":"The high temperature Ising model on the triangular lattice is a critical percolation model","attachmentId":79372558,"attachmentType":"pdf","work_url":"https://www.academia.edu/69177743/The_high_temperature_Ising_model_on_the_triangular_lattice_is_a_critical_percolation_model","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/69177743/The_high_temperature_Ising_model_on_the_triangular_lattice_is_a_critical_percolation_model"><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="22" data-entity-id="41711025" 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/41711025/Critical_behavior_of_the_Ising_model_on_non_local_directed_Small_World_networks">Critical behavior of the Ising model on non-local directed Small- World networks</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="2594470" href="https://independent.academia.edu/iosrjournals">IOSR Journals</a></div><p class="ds-related-work--metadata ds2-5-body-xs">IOSR Journals , 2019</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":"Critical behavior of the Ising model on non-local directed Small- World networks","attachmentId":61869932,"attachmentType":"pdf","work_url":"https://www.academia.edu/41711025/Critical_behavior_of_the_Ising_model_on_non_local_directed_Small_World_networks","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/41711025/Critical_behavior_of_the_Ising_model_on_non_local_directed_Small_World_networks"><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="23" data-entity-id="57691354" 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/57691354/Location_of_the_Lee_Yang_zeros_and_absence_of_phase_transitions_in_some_Ising_spin_systems">Location of the Lee-Yang zeros and absence of phase transitions in some Ising spin systems</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="40985101" href="https://independent.academia.edu/DavidRuelle">David Ruelle</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Mathematical Physics, 2012</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":"Location of the Lee-Yang zeros and absence of phase transitions in some Ising spin systems","attachmentId":72468683,"attachmentType":"pdf","work_url":"https://www.academia.edu/57691354/Location_of_the_Lee_Yang_zeros_and_absence_of_phase_transitions_in_some_Ising_spin_systems","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/57691354/Location_of_the_Lee_Yang_zeros_and_absence_of_phase_transitions_in_some_Ising_spin_systems"><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="24" data-entity-id="118041850" 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/118041850/Lee_Yang_zeros_and_the_Ising_model_on_the_Sierpinski_gasket">Lee-Yang zeros and the Ising model on the Sierpinski gasket</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="34786075" href="https://independent.academia.edu/DavideCassi">Davide Cassi</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Physics A: Mathematical and General, 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":"Lee-Yang zeros and the Ising model on the Sierpinski gasket","attachmentId":113759543,"attachmentType":"pdf","work_url":"https://www.academia.edu/118041850/Lee_Yang_zeros_and_the_Ising_model_on_the_Sierpinski_gasket","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/118041850/Lee_Yang_zeros_and_the_Ising_model_on_the_Sierpinski_gasket"><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></div><div class="footer--content"><ul class="footer--main-links hide-on-mobile"><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/hiring"><svg style="width: 13px; height: 13px; position: relative; bottom: -1px;" 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 href="https://support.academia.edu/"><svg style="width: 12px; height: 12px; position: relative; bottom: -1px;" 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--research-interests"><li>Find new research papers in:</li><li><a href="https://www.academia.edu/Documents/in/Physics">Physics</a></li><li><a href="https://www.academia.edu/Documents/in/Chemistry">Chemistry</a></li><li><a href="https://www.academia.edu/Documents/in/Biology">Biology</a></li><li><a href="https://www.academia.edu/Documents/in/Health_Sciences">Health Sciences</a></li><li><a href="https://www.academia.edu/Documents/in/Ecology">Ecology</a></li><li><a href="https://www.academia.edu/Documents/in/Earth_Sciences">Earth Sciences</a></li><li><a href="https://www.academia.edu/Documents/in/Cognitive_Science">Cognitive Science</a></li><li><a href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a></li><li><a href="https://www.academia.edu/Documents/in/Computer_Science">Computer Science</a></li></ul><ul class="footer--legal-links hide-on-mobile"><li><a href="https://www.academia.edu/terms">Terms</a></li><li><a href="https://www.academia.edu/privacy">Privacy</a></li><li><a href="https://www.academia.edu/copyright">Copyright</a></li><li>Academia ©2024</li></ul></div> </body> </html>