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<!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="Vt49yfw723wy987SPWgDva7es5mNlb6jkaZ1jiXGFQt6vUKXmFFnKbiDDRdYsIlEuqShf8q_iCOYKxV9u2sIbQ" /> <meta name="citation_title" content="Boundary Condition Dependence of Cluster Size Ratios in Random Percolation" /> <meta name="citation_publication_date" content="2000/01/01" /> <meta name="citation_journal_title" content="International Journal of Modern Physics C" /> <meta name="citation_author" content="Crisógono Rodrigues da Silva" /> <meta name="twitter:card" content="summary" /> <meta name="twitter:url" content="https://www.academia.edu/81833200/Boundary_Condition_Dependence_of_Cluster_Size_Ratios_in_Random_Percolation" /> <meta name="twitter:title" content="Boundary Condition Dependence of Cluster Size Ratios in Random Percolation" /> <meta name="twitter:description" content="We study the ratio of the number of sites in the largest and second largest clusters in random percolation. Using the scaling hypothesis that the ratio / of the mean cluster sizes M1 and M2 scales as f ((p - pc) L1/ν), we employ finite-size scaling" /> <meta name="twitter:image" content="https://0.academia-photos.com/115767637/51063012/39119090/s200_cris_gono_rodrigues_da.silva.jpeg" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/81833200/Boundary_Condition_Dependence_of_Cluster_Size_Ratios_in_Random_Percolation" /> <meta property="og:title" content="Boundary Condition Dependence of Cluster Size Ratios in Random Percolation" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="We study the ratio of the number of sites in the largest and second largest clusters in random percolation. Using the scaling hypothesis that the ratio / of the mean cluster sizes M1 and M2 scales as f ((p - pc) L1/ν), we employ finite-size scaling" /> <meta property="article:author" content="https://independent.academia.edu/Cris%C3%B3gonoRodriguesdaSilva" /> <meta name="description" content="We study the ratio of the number of sites in the largest and second largest clusters in random percolation. Using the scaling hypothesis that the ratio / of the mean cluster sizes M1 and M2 scales as f ((p - pc) L1/ν), we employ finite-size scaling" /> <title>(PDF) Boundary Condition Dependence of Cluster Size Ratios in Random Percolation</title> <link rel="canonical" href="https://www.academia.edu/81833200/Boundary_Condition_Dependence_of_Cluster_Size_Ratios_in_Random_Percolation" /> <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 = 'b092bf3a3df71cf13feee7c143e83a57eb6b94fb'; 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(1739818804000); window.Aedu.timeDifference = new Date().getTime() - 1739818804000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"We study the ratio of the number of sites in the largest and second largest clusters in random percolation. Using the scaling hypothesis that the ratio / of the mean cluster sizes M1 and M2 scales as f ((p - pc) L1/ν), we employ finite-size scaling analysis to find that / is nonuniversal with respect to the boundary conditions imposed. The mean of the ratios behaves similarly although with a distinct critical value reflecting the relevance of mass fluctuations at the percolation threshold. These zero exponent ratios also allow for reliable estimates of the critical parameters at percolation from relatively small lattices.","author":[{"@context":"https://schema.org","@type":"Person","name":"Crisógono Rodrigues da Silva","url":"https://independent.academia.edu/Cris%C3%B3gonoRodriguesdaSilva"}],"contributor":[],"dateCreated":"2022-06-19","dateModified":"2023-08-06","datePublished":"2000-01-01","headline":"Boundary Condition Dependence of Cluster Size Ratios in Random Percolation","image":"https://attachments.academia-assets.com/87739904/thumbnails/1.jpg","inLanguage":"en","keywords":["Mathematical Sciences","Physical sciences"],"publication":"International Journal of Modern Physics C","publisher":{"@context":"https://schema.org","@type":"Organization","name":"World Scientific Pub Co Pte 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window.loswp.work = {"work":{"id":81833200,"created_at":"2022-06-19T09:52:13.989-07:00","from_world_paper_id":208793467,"updated_at":"2024-06-20T07:11:01.755-07:00","_data":{"abstract":"We study the ratio of the number of sites in the largest and second largest clusters in random percolation. Using the scaling hypothesis that the ratio / of the mean cluster sizes M1 and M2 scales as f ((p - pc) L1/ν), we employ finite-size scaling analysis to find that / is nonuniversal with respect to the boundary conditions imposed. The mean of the ratios behaves similarly although with a distinct critical value reflecting the relevance of mass fluctuations at the percolation threshold. These zero exponent ratios also allow for reliable estimates of the critical parameters at percolation from relatively small lattices.","publisher":"World Scientific Pub Co Pte Lt","publication_date":"2000,,","publication_name":"International Journal of Modern Physics C"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Boundary Condition Dependence of Cluster Size Ratios in Random Percolation","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [115767637]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "control"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</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":87739904,"attachmentType":"pdf"}"><img alt="First page of “Boundary Condition Dependence of Cluster Size Ratios in Random Percolation”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/87739904/mini_magick20220619-17115-6sfpjs.png?1655657641" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.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">Boundary Condition Dependence of Cluster Size Ratios in Random Percolation</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="115767637" href="https://independent.academia.edu/Cris%C3%B3gonoRodriguesdaSilva"><img alt="Profile image of Crisógono Rodrigues da Silva" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/115767637/51063012/39119090/s65_cris_gono_rodrigues_da.silva.jpeg" />Crisógono Rodrigues da Silva</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2000, International Journal of Modern Physics C</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">5 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 81833200; 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Using the scaling hypothesis that the ratio / of the mean cluster sizes M1 and M2 scales as f ((p - pc) L1/ν), we employ finite-size scaling analysis to find that / is nonuniversal with respect to the boundary conditions imposed. The mean of the ratios behaves similarly although with a distinct critical value reflecting the relevance of mass fluctuations at the percolation threshold. These zero exponent ratios also allow for reliable estimates of the critical parameters at percolation from relatively small lattices.</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":87739904,"attachmentType":"pdf","workUrl":"https://www.academia.edu/81833200/Boundary_Condition_Dependence_of_Cluster_Size_Ratios_in_Random_Percolation"}">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":87739904,"attachmentType":"pdf","workUrl":"https://www.academia.edu/81833200/Boundary_Condition_Dependence_of_Cluster_Size_Ratios_in_Random_Percolation"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div><div class="ds-signup-banner-trigger-container"><div class="ds-signup-banner-trigger ds-signup-banner-trigger-control"></div></div><div class="ds-signup-banner ds-signup-banner-control"><div id="ds-signup-banner-close-button"><button class="ds2-5-button ds2-5-button--secondary ds2-5-button--inverse"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">close</span></button></div><div class="ds-signup-banner-ctas"><img src="//a.academia-assets.com/images/academia-logo-capital-white.svg" /><h4 class="ds2-5-heading-serif-sm">Sign up for access to the world's latest research</h4><button class="ds2-5-button ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{"location":"signup-banner"}">Sign up for free<span class="material-symbols-outlined" style="font-size: 20px" translate="no">arrow_forward</span></button></div><div class="ds-signup-banner-divider"></div><div class="ds-signup-banner-reasons"><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Get notified about relevant papers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Save papers to use in your research</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Join the discussion with peers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Track your impact</span></div></div></div><script>(() => { // Set up signup banner show/hide behavior: // 1. 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This analysis leads to estimates of the critical exponent and the ratio of critical exponents ␤/. ͓S0556-2813͑97͒02703-9͔</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":"Scaling behavior in very small percolation lattices","attachmentId":92802563,"attachmentType":"pdf","work_url":"https://www.academia.edu/88909428/Scaling_behavior_in_very_small_percolation_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-wsj-grid-card-view-pdf" href="https://www.academia.edu/88909428/Scaling_behavior_in_very_small_percolation_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-wsj-grid-card" data-collection-position="1" data-entity-id="20594466" 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/20594466/Nature_of_the_largest_cluster_size_distribution_at_the_percolation_threshold">Nature of the largest cluster size distribution at the percolation threshold</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">Journal of Physics A: Mathematical and General, 2001</p><p class="ds-related-work--abstract ds2-5-body-sm">Two distinct distribution functions Psp(m) and Pns(m) of the scaled largest cluster sizes m are obtained at the percolation threshold by numerical simulations, depending on the condition whether the lattice is actually spanned or not. With R(pc) the spanning probability, the total distribution of the largest cluster is given by Ptot(m) = R(pc)Psp(m) + (1 − R(pc))Pns(m). The three distributions apparently have similar forms in three and four dimensions while in two dimensions, Ptot(m) does not follow a familiar form. By studying the first and second cumulants of the distribution functions, the different behaviour of Ptot(m) in different dimensions may be quantified.</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":"Nature of the largest cluster size distribution at the percolation threshold","attachmentId":41455822,"attachmentType":"pdf","work_url":"https://www.academia.edu/20594466/Nature_of_the_largest_cluster_size_distribution_at_the_percolation_threshold","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/20594466/Nature_of_the_largest_cluster_size_distribution_at_the_percolation_threshold"><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="109669374" 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/109669374/Percolation_of_randomly_distributed_growing_clusters_Finite_size_scaling_and_critical_exponents_for_the_square_lattice">Percolation of randomly distributed growing clusters: Finite-size scaling and critical exponents for the square lattice</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="274688056" href="https://independent.academia.edu/MMaragakis">Michael Maragakis</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2010</p><p class="ds-related-work--abstract ds2-5-body-sm">We study the percolation properties of the growing clusters model. In this model, a number of seeds placed on random locations on a lattice are allowed to grow with a constant velocity to form clusters. When two or more clusters eventually touch each other they immediately stop their growth. The model exhibits a discontinuous transition for very low values of the seed concentration p and a second, non-trivial continuous phase transition for intermediate p values. Here we study in detail this continuous transition that separates a phase of finite clusters from a phase characterized by the presence of a giant component. Using finite size scaling and large scale Monte Carlo simulations we determine the value of the percolation threshold where the giant component first appears, and the critical exponents that characterize the transition. We find that the transition belongs to a different universality class from the standard percolation transition.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Percolation of randomly distributed growing clusters: Finite-size scaling and critical exponents for the square lattice","attachmentId":107725227,"attachmentType":"pdf","work_url":"https://www.academia.edu/109669374/Percolation_of_randomly_distributed_growing_clusters_Finite_size_scaling_and_critical_exponents_for_the_square_lattice","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/109669374/Percolation_of_randomly_distributed_growing_clusters_Finite_size_scaling_and_critical_exponents_for_the_square_lattice"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="3" data-entity-id="32317020" 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/32317020/The_Birth_of_the_Infinite_Cluster_Finite_Size_Scaling_in_Percolation">The Birth of the Infinite Cluster: Finite-Size Scaling in Percolation</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="62648080" href="https://independent.academia.edu/ChristianBorgs">Christian Borgs</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Rsa, 1997</p><p class="ds-related-work--abstract ds2-5-body-sm">We address the question of finite-size scaling in percolation by studying bond percolation in a finite box of side length n, both in two and in higher dimensions. In dimension d = 2, we obtain a complete characterization of finite-size scaling. In dimensions d > 2, we establish the same results under a set of hypotheses related to so-called scaling and hyperscaling postulates which are widely believed to hold up to d = 6.</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 Birth of the Infinite Cluster: Finite-Size Scaling in Percolation","attachmentId":52528983,"attachmentType":"pdf","work_url":"https://www.academia.edu/32317020/The_Birth_of_the_Infinite_Cluster_Finite_Size_Scaling_in_Percolation","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/32317020/The_Birth_of_the_Infinite_Cluster_Finite_Size_Scaling_in_Percolation"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="72797299" 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/72797299/Planar_Critical_Percolation_Large_clusters_and_Scaling_limits">Planar Critical Percolation: Large clusters and Scaling limits</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="102711148" href="https://independent.academia.edu/ReneConijn">Rene Conijn</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2015</p><p class="ds-related-work--abstract ds2-5-body-sm">This chapter is based on [10] with Rob van den Berg. We consider (near-)critical percolation on the square lattice. Let M n be the size of the largest open cluster contained in the box [−n, n] 2 , and let π(n) be the probability that there is an open path from O to the boundary of the box. It is well-known (see [17]) that for all 0 < a < b the probability that M n is smaller than an 2 π(n) and the probability that M n is larger than bn 2 π(n) are bounded away from 0 as n → ∞. It is a natural question, which arises for instance in the study of so-called frozenpercolation processes, if a similar result holds for the probability that M n is between an 2 π(n) and bn 2 π(n). By a suitable partition of the box, and a careful construction involving the building blocks, we show that the answer to this question is armative. The`sublinearity' of 1/π(n) appears to be essential for the argument. percolation and FK-Ising This chapter is based on [20] with Federico Camia and Demeter Kiss. Under some general assumptions we construct the scaling limit of open clusters and their associated counting measures in a class of two-dimensional percolation models. Our results apply, in particular, to critical Bernoulli site percolation on the triangular lattice. We also provide conditional results for the critical FK-Ising model on the square lattice. Fundamental properties of the scaling limit, such as conformal covariance, are explored. Applications such as the scaling limit of the largest cluster in a bounded domain and a geometric representation of the magnetization eld for the critical Ising model are presented.</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":"Planar Critical Percolation: Large clusters and Scaling limits","attachmentId":81581233,"attachmentType":"pdf","work_url":"https://www.academia.edu/72797299/Planar_Critical_Percolation_Large_clusters_and_Scaling_limits","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/72797299/Planar_Critical_Percolation_Large_clusters_and_Scaling_limits"><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="20594536" 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/20594536/Probability_Distribution_and_Sizes_of_Spanning_Clusters_at_the_Percolation_Thresholds">Probability Distribution and Sizes of Spanning Clusters at the Percolation Thresholds</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">International Journal of Modern Physics C, 1997</p><p class="ds-related-work--abstract ds2-5-body-sm">For random percolation at p c , the probability distribution P (n) of the number of spanning clusters (n) has been studied in large scale simulations. The results are compatible with P (n) ∼ exp(−an 2 ) for all dimensions. We also study the variation of the average size (mass) of the spanning clusters when there are more than one spanning cluster. While the average size of the spanning clusters scales as usual with L D where D = d − β/ν for any number of clusters, it shows a smooth decrease as the number of spanning clusters increases.</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":"Probability Distribution and Sizes of Spanning Clusters at the Percolation Thresholds","attachmentId":41455788,"attachmentType":"pdf","work_url":"https://www.academia.edu/20594536/Probability_Distribution_and_Sizes_of_Spanning_Clusters_at_the_Percolation_Thresholds","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/20594536/Probability_Distribution_and_Sizes_of_Spanning_Clusters_at_the_Percolation_Thresholds"><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="57259212" 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/57259212/Critical_Behavior_in_Percolation_Processes">Critical Behavior in Percolation Processes</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="39346513" href="https://independent.academia.edu/RuddWalter">Walter Rudd</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review B</p><p class="ds-related-work--abstract ds2-5-body-sm">The Monte Carlo percolation-probability data of Frisch et al. are analyzed under the assumption that R(p), the probability that a given site (bond) belongs to an infinite cluster as a function of the probability p of site (bond) occupation, has the asymptotic behavior R(p) (p-p~)~, as p p~, where p~i s the critical percolation probability. The estimated values of P for various lattices are tabulated.</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 in Percolation Processes","attachmentId":72245257,"attachmentType":"pdf","work_url":"https://www.academia.edu/57259212/Critical_Behavior_in_Percolation_Processes","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/57259212/Critical_Behavior_in_Percolation_Processes"><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="26581783" 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/26581783/Monte_Carlo_experiments_on_cluster_size_distribution_in_percolation">Monte Carlo experiments on cluster size distribution in percolation</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="50563072" href="https://independent.academia.edu/JHoshen">J. Hoshen</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Physics A: Mathematical and General, 1979</p><p class="ds-related-work--abstract ds2-5-body-sm">Cluster statistics in two-and three-dimensional site percolation problems are derived here by Monte Carlo methods. The average number n, of percolation clusters with s occupied sites each is calculated by up to 19 runs on a 4000 X 4000 triangular lattice near p c .</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Monte Carlo experiments on cluster size distribution in percolation","attachmentId":46873148,"attachmentType":"pdf","work_url":"https://www.academia.edu/26581783/Monte_Carlo_experiments_on_cluster_size_distribution_in_percolation","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/26581783/Monte_Carlo_experiments_on_cluster_size_distribution_in_percolation"><span class="ds2-5-text-link__content">View PDF</span><span 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="5447429" 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/5447429/Monte_Carlo_and_Series_Study_of_Corrections_to_Scaling_in_Two_Dimensional_Percolation">Monte Carlo and Series Study of Corrections to Scaling in Two-Dimensional Percolation</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="7630487" href="https://bu.academia.edu/EugeneStanley">Eugene Stanley</a></div><p class="ds-related-work--abstract ds2-5-body-sm">Corrections to scaling for percolation cluster numbers in two dimensions are studied by Monte Carlo simulations of very large systems (up to 17 X lo9 lattice sites) and by series analysis. Both series and Monte Carlo work suggests that the value of the correction-to-scaling exponent is slightly lower at the percolation threshold than away from it. Moreover, the corrections to scaling observed at pc (a ~0 . 6 4 ) might be due to the mixing of scaling fields rather than to the irrelevant scaling fields. The Monte Carlo results are compatible with finite-size scaling, and finite-size scaling corrections are estimated. Technical problems associated with Monte Carlo simulation of very large systems are discussed in an appendix.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Monte Carlo and Series Study of Corrections to Scaling in Two-Dimensional Percolation","attachmentId":32570702,"attachmentType":"pdf","work_url":"https://www.academia.edu/5447429/Monte_Carlo_and_Series_Study_of_Corrections_to_Scaling_in_Two_Dimensional_Percolation","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/5447429/Monte_Carlo_and_Series_Study_of_Corrections_to_Scaling_in_Two_Dimensional_Percolation"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="85156478" 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/85156478/Universal_features_of_cluster_numbers_in_percolation">Universal features of cluster numbers in percolation</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="160629230" href="https://sfb-trr-62.academia.edu/StephanMertens">Stephan Mertens</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2017</p><p class="ds-related-work--abstract ds2-5-body-sm">The number of clusters per site n(p) in percolation at the critical point p = p c is not itself a universal quantity-it depends upon the lattice and percolation type (site or bond). However, many of its properties, including finite-size corrections, scaling behavior with p, and amplitude ratios, show various degrees of universal behavior. Some of these are universal in the sense that the behavior depends upon the shape of the system, but not lattice type. Here, we elucidate the various levels of universality for elements of n(p) both theoretically and by carrying out extensive studies on several two-and three-dimensional systems, by high-order series analysis, Monte-Carlo simulation, and exact enumeration. We find many new results, including precise values for n(p c) for several systems, a clear demonstration of the singularity in n (p), and metric scale factors. We make use of the matching polynomial of Sykes and Essam to find exact relations between properties for lattices and matching lattices. We propose a criterion for an absolute metric factor b based upon the singular behavior of the scaling function, rather than a relative definition of the metric that has previously been used.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Universal features of cluster numbers in percolation","attachmentId":89942060,"attachmentType":"pdf","work_url":"https://www.academia.edu/85156478/Universal_features_of_cluster_numbers_in_percolation","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/85156478/Universal_features_of_cluster_numbers_in_percolation"><span class="ds2-5-text-link__content">View 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":87739904,"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":87739904,"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_87739904" 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="54335511" 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/54335511/The_gaps_between_the_sizes_of_large_clusters_in_2D_critical_percolation">The gaps between the sizes of large clusters in 2D critical percolation</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="102711148" href="https://independent.academia.edu/ReneConijn">Rene Conijn</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Electronic Communications in Probability, 2013</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 gaps between the sizes of large clusters in 2D critical percolation","attachmentId":70749541,"attachmentType":"pdf","work_url":"https://www.academia.edu/54335511/The_gaps_between_the_sizes_of_large_clusters_in_2D_critical_percolation","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/54335511/The_gaps_between_the_sizes_of_large_clusters_in_2D_critical_percolation"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="1" data-entity-id="75159609" 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/75159609/Universal_scaling_functions_for_bond_percolation_on_planar_random_and_square_lattices_with_multiple_percolating_clusters">Universal scaling functions for bond percolation on planar random and square lattices with multiple percolating clusters</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="59280610" href="https://independent.academia.edu/SimonLin23">Simon Lin</a></div><p class="ds-related-work--metadata ds2-5-body-xs">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":"Universal scaling functions for bond percolation on planar random and square lattices with multiple percolating clusters","attachmentId":83041011,"attachmentType":"pdf","work_url":"https://www.academia.edu/75159609/Universal_scaling_functions_for_bond_percolation_on_planar_random_and_square_lattices_with_multiple_percolating_clusters","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/75159609/Universal_scaling_functions_for_bond_percolation_on_planar_random_and_square_lattices_with_multiple_percolating_clusters"><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 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Overview","attachmentId":79372543,"attachmentType":"pdf","work_url":"https://www.academia.edu/69177693/Scaling_Limits_of_Two_Dimensional_Percolation_an_Overview","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/69177693/Scaling_Limits_of_Two_Dimensional_Percolation_an_Overview"><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="54335771" 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/54335771/On_the_size_of_the_largest_cluster_in_2D_critical_percolation">On the size of the largest cluster in 2D critical percolation</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="102711148" href="https://independent.academia.edu/ReneConijn">Rene Conijn</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Electronic Communications in Probability, 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":"On the size of the largest cluster in 2D critical percolation","attachmentId":70749697,"attachmentType":"pdf","work_url":"https://www.academia.edu/54335771/On_the_size_of_the_largest_cluster_in_2D_critical_percolation","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/54335771/On_the_size_of_the_largest_cluster_in_2D_critical_percolation"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="4" data-entity-id="50621707" 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/50621707/Directed_compact_percolation_cluster_size_and_hyperscaling">Directed compact percolation: cluster size and hyperscaling</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="57543588" href="https://royalholloway.academia.edu/JohnEssam">John Essam</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Physics A: Mathematical and General, 1989</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":"Directed compact percolation: cluster size and hyperscaling","attachmentId":68535119,"attachmentType":"pdf","work_url":"https://www.academia.edu/50621707/Directed_compact_percolation_cluster_size_and_hyperscaling","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/50621707/Directed_compact_percolation_cluster_size_and_hyperscaling"><span 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data-author-id="52848710" href="https://independent.academia.edu/FedericoCamia">Federico Camia</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Statistical Physics, 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":"The Scaling Limit Geometry of Near-Critical 2D Percolation","attachmentId":79372651,"attachmentType":"pdf","work_url":"https://www.academia.edu/69177710/The_Scaling_Limit_Geometry_of_Near_Critical_2D_Percolation","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/69177710/The_Scaling_Limit_Geometry_of_Near_Critical_2D_Percolation"><span 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