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Theory Related Fields 139" /> <meta name="citation_publication_date" content="2007/01/01" /> <meta name="citation_author" content="Federico Camia" /> <meta name="twitter:card" content="summary" /> <meta name="twitter:url" content="https://www.academia.edu/69177770/Critical_percolation_exploration_path_and_SLE6_a_proof_of_convergence_Probab_Theory_Related_Fields_139" /> <meta name="twitter:title" content="Critical percolation exploration path and SLE6: a proof of convergence. Probab. Theory Related Fields 139" /> <meta name="twitter:description" content="It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE6. We provide here a detailed proof, which relies on Smirnov’s theorem that" /> <meta name="twitter:image" content="http://a.academia-assets.com/images/twitter-card.jpeg" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/69177770/Critical_percolation_exploration_path_and_SLE6_a_proof_of_convergence_Probab_Theory_Related_Fields_139" /> <meta property="og:title" content="Critical percolation exploration path and SLE6: a proof of convergence. Probab. Theory Related Fields 139" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE6. We provide here a detailed proof, which relies on Smirnov’s theorem that" /> <meta property="article:author" content="https://independent.academia.edu/FedericoCamia" /> <meta name="description" content="It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE6. We provide here a detailed proof, which relies on Smirnov’s theorem that" /> <title>(PDF) Critical percolation exploration path and SLE6: a proof of convergence. Probab. 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We provide here a detailed proof, which relies on Smirnov’s theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy’s formula). The version of convergence to SLE6 that we prove suffices for the Smirnov-Werner derivation of certain critical percolation crossing exponents and for our analysis of the critical percolation full scaling limit as a process of continuum nonsimple loops.","author":[{"@context":"https://schema.org","@type":"Person","name":"Federico Camia","url":"https://independent.academia.edu/FedericoCamia"}],"contributor":[],"dateCreated":"2022-01-22","dateModified":"2022-01-22","datePublished":"2007-01-01","headline":"Critical percolation exploration path and SLE6: a proof of convergence. Probab. 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We provide here a detailed proof, which relies on Smirnov’s theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy’s formula). The version of convergence to SLE6 that we prove suffices for the Smirnov-Werner derivation of certain critical percolation crossing exponents and for our analysis of the critical percolation full scaling limit as a process of continuum nonsimple loops.","publication_date":"2007,,"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Critical percolation exploration path and SLE6: a proof of convergence. Probab. Theory Related Fields 139","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [52848710]; 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":79372677,"attachmentType":"pdf"}"><img alt="First page of “Critical percolation exploration path and SLE6: a proof of convergence. Probab. Theory Related Fields 139”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/79372677/mini_magick20220122-23764-1dln1eu.png?1642871794" /><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">Critical percolation exploration path and SLE6: a proof of convergence. Probab. Theory Related Fields 139</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="52848710" href="https://independent.academia.edu/FedericoCamia"><img alt="Profile image of Federico Camia" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Federico Camia</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2007</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">47 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 = 69177770; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=69177770"; const getWorkViews = async (workId) => { const response = await fetch(worksViewsPath); if (!response.ok) { throw new Error('Failed to load work views'); } const data = await response.json(); return data.views[workId]; }; // Get the view count for the work - we send this immediately rather than waiting for // the DOM to load, so it can be available as soon as possible (but without holding up // the backend or other resource requests, because it's a bit expensive and not critical). const viewCount = await getWorkViews(workId); const updateViewCount = (viewCount) => { try { const viewCountNumber = parseInt(viewCount, 10); if (viewCountNumber === 0) { // Remove the whole views element if there are zero views. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); return; } const commaizedViewCount = viewCountNumber.toLocaleString(); const viewCountBody = document.getElementById('work-metadata-view-count'); if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE6. We provide here a detailed proof, which relies on Smirnov’s theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy’s formula). The version of convergence to SLE6 that we prove suffices for the Smirnov-Werner derivation of certain critical percolation crossing exponents and for our analysis of the critical percolation full scaling limit as a process of continuum nonsimple loops.</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":79372677,"attachmentType":"pdf","workUrl":"https://www.academia.edu/69177770/Critical_percolation_exploration_path_and_SLE6_a_proof_of_convergence_Probab_Theory_Related_Fields_139"}">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":79372677,"attachmentType":"pdf","workUrl":"https://www.academia.edu/69177770/Critical_percolation_exploration_path_and_SLE6_a_proof_of_convergence_Probab_Theory_Related_Fields_139"}"><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" data-impression-entity-id="69177770" data-impression-entity-type="2" data-impression-source="signup-banner"><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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We provide here a detailed proof, which relies on Smirnov&#x27;s theorem that crossing ...</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 percolation exploration path and SLE 6: a proof of convergence","attachmentId":79372646,"attachmentType":"pdf","work_url":"https://www.academia.edu/69177698/Critical_percolation_exploration_path_and_SLE_6_a_proof_of_convergence","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/69177698/Critical_percolation_exploration_path_and_SLE_6_a_proof_of_convergence"><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="79596871" 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/79596871/Continuum_Nonsimple_Loops_and_2D_Critical_Percolation">Continuum Nonsimple Loops and 2D Critical 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="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, 2004</p><p class="ds-related-work--abstract ds2-5-body-sm">Substantial progress has been made in recent years on the 2D critical percolation scaling limit and its conformal invariance properties. In particular, chordal SLE 6 (the Stochastic Löwner Evolution with parameter κ = 6) was, in the work of Schramm and of Smirnov, identified as the scaling limit of the critical percolation "exploration process." In this paper we use that and other results to construct what we argue is the full scaling limit of the collection of all closed contours surrounding the critical percolation clusters on the 2D triangular lattice. This random process or gas of continuum nonsimple loops in R 2 is constructed inductively by repeated use of chordal SLE 6. These loops do not cross but do touch each other-indeed, any two loops are connected by a finite "path" of touching loops.</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":"Continuum Nonsimple Loops and 2D Critical Percolation","attachmentId":86257458,"attachmentType":"pdf","work_url":"https://www.academia.edu/79596871/Continuum_Nonsimple_Loops_and_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-wsj-grid-card-view-pdf" href="https://www.academia.edu/79596871/Continuum_Nonsimple_Loops_and_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-wsj-grid-card" data-collection-position="2" data-entity-id="69177699" 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/69177699/The_full_scaling_limit_of_two_dimensional_critical_percolation">The full scaling limit of two-dimensional critical 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="52848710" href="https://independent.academia.edu/FedericoCamia">Federico Camia</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Arxiv preprint math/0504036, 2005</p><p class="ds-related-work--abstract ds2-5-body-sm">Abstract: We use SLE (6) paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice--that is, the scaling limit of the set of all interfaces ...</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 full scaling limit of two-dimensional critical percolation","attachmentId":79372644,"attachmentType":"pdf","work_url":"https://www.academia.edu/69177699/The_full_scaling_limit_of_two_dimensional_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-wsj-grid-card-view-pdf" href="https://www.academia.edu/69177699/The_full_scaling_limit_of_two_dimensional_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-wsj-grid-card" data-collection-position="3" data-entity-id="69177697" 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/69177697/Two_Dimensional_Critical_Percolation_The_Full_Scaling_Limit">Two-Dimensional Critical Percolation: The Full Scaling Limit</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="52848710" href="https://independent.academia.edu/FedericoCamia">Federico Camia</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Communications in Mathematical Physics, 2006</p><p class="ds-related-work--abstract ds2-5-body-sm">We use SLE 6 paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice-that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved.</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":"Two-Dimensional Critical Percolation: The Full Scaling Limit","attachmentId":79372640,"attachmentType":"pdf","work_url":"https://www.academia.edu/69177697/Two_Dimensional_Critical_Percolation_The_Full_Scaling_Limit","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/69177697/Two_Dimensional_Critical_Percolation_The_Full_Scaling_Limit"><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="69177700" 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/69177700/The_scaling_limit_geometry_of_near_critical_2D_percolation">The scaling limit geometry of near-critical 2D 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="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><p class="ds-related-work--abstract ds2-5-body-sm">We analyze the geometry of scaling limits of near-critical 2D percolation, ie, for p= p c+ λδ 1/ν, with ν= 4/3, as the lattice spacing δ→ 0. Our proposed framework extends previous analyses for p= pc, based on SLE 6. It combines the continuum nonsimple loop ...</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":79372641,"attachmentType":"pdf","work_url":"https://www.academia.edu/69177700/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-wsj-grid-card-view-pdf" href="https://www.academia.edu/69177700/The_scaling_limit_geometry_of_near_critical_2D_percolation"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" 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="6" data-entity-id="69177693" 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/69177693/Scaling_Limits_of_Two_Dimensional_Percolation_an_Overview">Scaling Limits of Two-Dimensional Percolation: an Overview</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="52848710" href="https://independent.academia.edu/FedericoCamia">Federico Camia</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Statistica Neerlandica, 2008</p><p class="ds-related-work--abstract ds2-5-body-sm">We present a review of the recent progress on percolation scaling limits in two dimensions. In particular, we will consider the convergence of critical crossing probabilities to Cardy&#x27;s formula and of the critical exploration path to chordal SLE(6), the full scaling limit of critical cluster boundaries, and near-critical scaling limits.</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 Limits of Two-Dimensional Percolation: an 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-wsj-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-wsj-grid-card" data-collection-position="7" data-entity-id="89261405" 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/89261405/Mean_field_critical_behaviour_for_percolation_in_high_dimensions">Mean-field critical behaviour for percolation in high dimensions</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="40413618" href="https://independent.academia.edu/TakashiHara1">Takashi Hara</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Communications in Mathematical Physics, 1990</p><p class="ds-related-work--abstract ds2-5-body-sm">The triangle condition for percolation states that ]Γ τ(0,x) τ(x,y) χ,y-τ (y, 0) is finite at the critical point, where τ (x, y) is the probability that the sites x and y are connected. We use an expansion related to the lace expansion for a self-avoiding walk to prove that the triangle condition is satisfied in two situations: (i) for nearest-neighbour independent bond percolation on thê /-dimensional hypercubic lattice, if d is sufficiently large, and (ii) in more than six dimensions for a class of "spread-out" models of independent bond percolation which are believed to be in the same universality class as the nearest-neighbour model. The class of models in (ii) includes the case where the bond occupation probability is constant for bonds of length less than some large number, and is zero otherwise. In the course of the proof an infrared bound is obtained. The triangle condition is known to imply that various critical exponents take their mean-field (Bethe lattice) values (y = β = 1, δ = Δ t = 2, ί ^ 2) and that the percolation density is continuous at the critical point. We also prove that v 2 = 1/2 in (i) and (ii), where v 2 is the critical exponent for the correlation length.</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":"Mean-field critical behaviour for percolation in high dimensions","attachmentId":93092161,"attachmentType":"pdf","work_url":"https://www.academia.edu/89261405/Mean_field_critical_behaviour_for_percolation_in_high_dimensions","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/89261405/Mean_field_critical_behaviour_for_percolation_in_high_dimensions"><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="89261403" 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/89261403/The_Self_Avoiding_Walk_and_Percolation_Critical_Points_in_High_Dimensions">The Self-Avoiding-Walk and Percolation Critical Points in High Dimensions</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="40413618" href="https://independent.academia.edu/TakashiHara1">Takashi Hara</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Combinatorics, Probability and Computing, 1995</p><p class="ds-related-work--abstract ds2-5-body-sm">We prove the existence of an asymptotic expansion in the inverse dimension, to all orders, for the connective constant for self-avoiding walks on ℤd. For the critical point, defined as the reciprocal of the connective constant, the coefficients of the expansion are computed through orderd−6, with a rigorous error bound of orderd−7Our method for computing terms in the expansion also applies to percolation, and for nearest-neighbour independent Bernoulli bond percolation on ℤdgives the 1/d-expansion for the critical point through orderd−3, with a rigorous error bound of orderd−4The method uses the lace expansion.</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 Self-Avoiding-Walk and Percolation Critical Points in High Dimensions","attachmentId":93092151,"attachmentType":"pdf","work_url":"https://www.academia.edu/89261403/The_Self_Avoiding_Walk_and_Percolation_Critical_Points_in_High_Dimensions","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/89261403/The_Self_Avoiding_Walk_and_Percolation_Critical_Points_in_High_Dimensions"><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="30144809" 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/30144809/Series_expansions_of_the_percolation_probability_on_the_directed_triangular_lattice">Series expansions of the percolation probability on the directed triangular 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="57364483" href="https://unimelb.academia.edu/IwanJensen">Iwan Jensen</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Physics A: Mathematical and General, 1996</p><p class="ds-related-work--abstract ds2-5-body-sm">We have derived long series expansions of the percolation probability for site, bond and site-bond percolation on the directed triangular lattice. For the bond problem we have extended the series from order 12 to 51 and for the site problem from order 12 to 35. For the site-bond problem, which has not been studied before, we have derived the series to order 32. Our estimates of the critical exponent β are in full agreement with results for similar problems on the square lattice, confirming expectations of universality. For the critical probability and exponent we find in the site case: q c = 0.4043528 ± 0.0000010 and β = 0.27645 ± 0.00010; in the bond case: q c = 0.52198 ± 0.00001 and β = 0.2769 ± 0.0010; and in the site-bond case: q c = 0.264173 ± 0.000003 and β = 0.2766 ± 0.0003. In addition we have obtained accurate estimates for the critical amplitudes. In all cases we find that the leading correction to scaling term is analytic, i.e., the confluent exponent ∆ = 1.</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":"Series expansions of the percolation probability on the directed triangular lattice","attachmentId":50602172,"attachmentType":"pdf","work_url":"https://www.academia.edu/30144809/Series_expansions_of_the_percolation_probability_on_the_directed_triangular_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/30144809/Series_expansions_of_the_percolation_probability_on_the_directed_triangular_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></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":79372677,"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":79372677,"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_79372677" 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="69177707" 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/69177707/Trivial_Critical_and_Near_critical_Scaling_Limits_of_Two_dimensional_Percolation">Trivial, Critical and Near-critical Scaling Limits of Two-dimensional 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="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, 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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/25600961/Critical_percolation_on_mesoscopic_triangulations"><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="121595013" 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/121595013/On_Crossing_Event_Formulas_in_Critical_Two_Dimensional_Percolation">On Crossing Event Formulas in Critical Two-Dimensional 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="307800349" href="https://independent.academia.edu/RobertSMaier">Robert S. Maier</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Statistical Physics, 2003</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 Crossing Event Formulas in Critical Two-Dimensional Percolation","attachmentId":116433199,"attachmentType":"pdf","work_url":"https://www.academia.edu/121595013/On_Crossing_Event_Formulas_in_Critical_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-related-work-grid-card-view-pdf" href="https://www.academia.edu/121595013/On_Crossing_Event_Formulas_in_Critical_Two_Dimensional_Percolation"><span class="ds2-5-text-link__content">View PDF</span><span 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percolation probabilities","attachmentId":66658679,"attachmentType":"pdf","work_url":"https://www.academia.edu/47702150/On_the_critical_percolation_probabilities","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/47702150/On_the_critical_percolation_probabilities"><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="3090564" 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/3090564/Critical_Behavior_for_an_Oriented_Percolation_with_Long_Range_Interactions_in_Dimension_d_2">Critical Behavior for an Oriented Percolation with Long-Range Interactions in Dimension d > 2</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="798955" href="https://taiwan.academia.edu/NarnRueihShieh">Narn-Rueih Shieh</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":"Critical Behavior for an Oriented Percolation with Long-Range Interactions in Dimension d \u003e 2","attachmentId":31017961,"attachmentType":"pdf","work_url":"https://www.academia.edu/3090564/Critical_Behavior_for_an_Oriented_Percolation_with_Long_Range_Interactions_in_Dimension_d_2","alternativeTracking":true}"><span class="material-symbols-outlined" 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class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32298185" href="https://huji.academia.edu/ScottKirkpatrick">Scott Kirkpatrick</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review Letters, 1976</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 Phenomena in Higher Dimensions: Approach to the Mean-Field Limit","attachmentId":120762024,"attachmentType":"pdf","work_url":"https://www.academia.edu/126959400/Percolation_Phenomena_in_Higher_Dimensions_Approach_to_the_Mean_Field_Limit","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" 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href="https://www.academia.edu/54335823/Factorization_Formulas_for_2D_Critical_Percolation_Revisited">Factorization Formulas for $2D$ Critical Percolation, Revisited</a><div class="ds-related-work--metadata"><a class="js-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">2015</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Factorization Formulas for $2D$ Critical Percolation, Revisited","attachmentId":70749716,"attachmentType":"pdf","work_url":"https://www.academia.edu/54335823/Factorization_Formulas_for_2D_Critical_Percolation_Revisited","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span 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href="https://independent.academia.edu/MichaelAizenman">Michael Aizenman</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 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":"Continuum Limits for Critical Percolation and Other Stochastic Geometric Models","attachmentId":113640020,"attachmentType":"pdf","work_url":"https://www.academia.edu/117780770/Continuum_Limits_for_Critical_Percolation_and_Other_Stochastic_Geometric_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-related-work-grid-card-view-pdf" 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