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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="ZtRgbsrxtK75LKkS8XkBoRDSeuwlQGzZbLMWcAXMDN0cofpTZnfbu0UzrfigAM9RbQjwHudwqVma8fn-DTS75A" /> <meta name="citation_title" content="On Crossing Event Formulas in Critical Two-Dimensional Percolation" /> <meta name="citation_publication_date" content="2003/01/01" /> <meta name="citation_journal_title" content="Journal of Statistical Physics" /> <meta name="citation_author" content="Robert S. 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These include Watts&#39;s formula for the horizontal-vertical crossing probability and Cardy&#39;s new formula for the" /> <meta name="twitter:image" content="https://0.academia-photos.com/307800349/152201299/141796070/s200_robert_s..maier.png" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/121595013/On_Crossing_Event_Formulas_in_Critical_Two_Dimensional_Percolation" /> <meta property="og:title" content="On Crossing Event Formulas in Critical Two-Dimensional Percolation" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="Several formulas for crossing functions arising in the continuum limit of critical two-dimensional percolation models are studied. These include Watts&#39;s formula for the horizontal-vertical crossing probability and Cardy&#39;s new formula for the" /> <meta property="article:author" content="https://independent.academia.edu/RobertSMaier" /> <meta name="description" content="Several formulas for crossing functions arising in the continuum limit of critical two-dimensional percolation models are studied. These include Watts&#39;s formula for the horizontal-vertical crossing probability and Cardy&#39;s new formula for the" /> <title>(PDF) On Crossing Event Formulas in Critical Two-Dimensional Percolation</title> <link rel="canonical" href="https://www.academia.edu/121595013/On_Crossing_Event_Formulas_in_Critical_Two_Dimensional_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 = '1e60a92a442ff83025cbe4f252857ee7c49c0bbe'; 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(1740573797000); window.Aedu.timeDifference = new Date().getTime() - 1740573797000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"Several formulas for crossing functions arising in the continuum limit of critical two-dimensional percolation models are studied. These include Watts\u0026amp;amp;#39;s formula for the horizontal-vertical crossing probability and Cardy\u0026amp;amp;#39;s new formula for the expected number of crossing clusters. It is shown that for lattices where conformal invariance holds, they simplify when the spatial domain is taken to be the interior of an equilateral triangle. The two crossing functions can be expressed in terms of an equianharmonic elliptic function with a triangular rotational symmetry. This suggests that rigorous proofs of Watts\u0026amp;amp;#39;s formula and Cardy\u0026amp;amp;#39;s new formula will be easiest to construct if the underlying lattice is triangular. The simplification in a triangular domain of Schramm\u0026amp;amp;#39;s “bulk Cardy\u0026amp;amp;#39;s formula” is also studied.","author":[{"@context":"https://schema.org","@type":"Person","name":"Robert S. Maier","url":"https://independent.academia.edu/RobertSMaier"}],"contributor":[],"dateCreated":"2024-06-28","dateModified":"2024-11-27","datePublished":"2003-01-01","headline":"On Crossing Event Formulas in Critical Two-Dimensional Percolation","image":"https://attachments.academia-assets.com/116433199/thumbnails/1.jpg","inLanguage":"en","keywords":["Mathematics","Mathematical Physics","Physics","Statistical Physics","Mathematical Sciences","Physical sciences","Special functions","Statistical","Hexagonal lattice","Equilateral triangle","Conformal Invariance","Conformal Symmetry"],"publication":"Journal of Statistical 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{"work":{"id":121595013,"created_at":"2024-06-28T10:34:38.559-07:00","from_world_paper_id":256756290,"updated_at":"2024-11-27T22:42:13.330-08:00","_data":{"abstract":"Several formulas for crossing functions arising in the continuum limit of critical two-dimensional percolation models are studied. These include Watts\u0026#39;s formula for the horizontal-vertical crossing probability and Cardy\u0026#39;s new formula for the expected number of crossing clusters. It is shown that for lattices where conformal invariance holds, they simplify when the spatial domain is taken to be the interior of an equilateral triangle. The two crossing functions can be expressed in terms of an equianharmonic elliptic function with a triangular rotational symmetry. This suggests that rigorous proofs of Watts\u0026#39;s formula and Cardy\u0026#39;s new formula will be easiest to construct if the underlying lattice is triangular. The simplification in a triangular domain of Schramm\u0026#39;s “bulk Cardy\u0026#39;s formula” is also studied.","ai_title_tag":"Crossing Formulas in Critical 2D Percolation Models","publication_date":"2003,,","publication_name":"Journal of Statistical Physics"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"On Crossing Event Formulas in Critical Two-Dimensional Percolation","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [307800349]; 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.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":116433199,"attachmentType":"pdf"}"><img alt="First page of “On Crossing Event Formulas in Critical Two-Dimensional Percolation”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/116433199/mini_magick20240801-1-cx5y6l.png?1722549442" /><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">On Crossing Event Formulas in Critical Two-Dimensional 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="307800349" href="https://independent.academia.edu/RobertSMaier"><img alt="Profile image of Robert S. Maier" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/307800349/152201299/141796070/s65_robert_s..maier.png" />Robert S. Maier</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2003, Journal of Statistical Physics</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">21 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 = 121595013; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=121595013"; 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">Several formulas for crossing functions arising in the continuum limit of critical two-dimensional percolation models are studied. These include Watts&#39;s formula for the horizontal-vertical crossing probability and Cardy&#39;s new formula for the expected number of crossing clusters. It is shown that for lattices where conformal invariance holds, they simplify when the spatial domain is taken to be the interior of an equilateral triangle. The two crossing functions can be expressed in terms of an equianharmonic elliptic function with a triangular rotational symmetry. This suggests that rigorous proofs of Watts&#39;s formula and Cardy&#39;s new formula will be easiest to construct if the underlying lattice is triangular. The simplification in a triangular domain of Schramm&#39;s “bulk Cardy&#39;s formula” is also studied.</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":116433199,"attachmentType":"pdf","workUrl":"https://www.academia.edu/121595013/On_Crossing_Event_Formulas_in_Critical_Two_Dimensional_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":116433199,"attachmentType":"pdf","workUrl":"https://www.academia.edu/121595013/On_Crossing_Event_Formulas_in_Critical_Two_Dimensional_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" data-impression-entity-id="121595013" 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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Let $u_1, u_2$ be two sites on the boundary and $w$ a site in the interior of the half-plane. It was predicted by Simmons, Kleban and Ziff in a paper from 2007 that the ratio $\mathbb{P}(nu_1 \leftrightarrow nu_2 \leftrightarrow nw)^{2}\,/\,\mathbb{P}(nu_1 \leftrightarrow nu_2)\cdot\mathbb{P}(nu_1 \leftrightarrow nw)\cdot\mathbb{P}(nu_2 \leftrightarrow nw)$ converges to $K_F$ as $n \to \infty$, where $x\leftrightarrow y$ denotes the event that $x$ and $y$ are in the same open cluster, and $K_F$ is an explicitly known constant. Beliaev and Izyurov proved in a paper in 2012 an analog of this factorization in the scaling limit. We prove, using their result and a generalized coupling argument, the earlier mentioned prediction. Furthermore we prove a factorization formula for the probability $\mathbb{P}(nu_2 \leftrightarrow [nu_1,nu_1+s];\, nw \leftrightarrow [nu_1,nu_1+s])$, where $s>0$.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Factorization Formulas for $2D$ Critical Percolation, Revisited","attachmentId":70749716,"attachmentType":"pdf","work_url":"https://www.academia.edu/54335823/Factorization_Formulas_for_2D_Critical_Percolation_Revisited","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/54335823/Factorization_Formulas_for_2D_Critical_Percolation_Revisited"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="2" 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. 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M. T. Watts derived in his paper [20] that in two dimensional critical percolation the crossing probability Π hv satisfies a fifth order differential equation which includes another one of third order whose independent solutions describe the physically relevant quantities 1, Π h , Π hv .</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":"Conformal Field Theory Properties of Two-Dimensional Percolation","attachmentId":32678318,"attachmentType":"pdf","work_url":"https://www.academia.edu/5600068/Conformal_Field_Theory_Properties_of_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/5600068/Conformal_Field_Theory_Properties_of_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="5" data-entity-id="14707169" 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/14707169/Universality_of_the_excess_number_of_clusters_and_the_crossing_probability_function_in_three_dimensional_percolation">Universality of the excess number of clusters and the crossing probability function in three-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="33660420" href="https://kcl.academia.edu/ChristianLorenz">Christian Lorenz</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Physics A: Mathematical and General, 1998</p><p class="ds-related-work--abstract ds2-5-body-sm">Extensive Monte-Carlo simulations were performed to evaluate the excess number of clusters and the crossing probability function for three-dimensional percolation on the simple cubic (s.c.), face-centered cubic (f.c.c.), and bodycentered cubic (b.c.c.) lattices. Systems L × L × L ′ with L ′ >> L were studied for both bond (s.c., f.c.c., b.c.c.) and site (f.c.c.) percolation. The excess number of clustersb per unit length was confirmed to be a universal quantity with a valueb ≈ 0.412. Likewise, the critical crossing probability in the L ′ direction, with periodic boundary conditions in the L × L plane, was found to follow a universal exponential decay as a function of r = L ′ /L for large r. Simulations were also carried out to find new precise values of the critical thresholds for site percolation on the f.c.c. and b.c.c. lattices, yielding</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":"Universality of the excess number of clusters and the crossing probability function in three-dimensional percolation","attachmentId":43956888,"attachmentType":"pdf","work_url":"https://www.academia.edu/14707169/Universality_of_the_excess_number_of_clusters_and_the_crossing_probability_function_in_three_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/14707169/Universality_of_the_excess_number_of_clusters_and_the_crossing_probability_function_in_three_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="6" data-entity-id="22624017" 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/22624017/Probability_of_incipient_spanning_clusters_in_critical_two_dimensional_percolation">Probability of incipient spanning clusters in critical 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="44196240" href="https://independent.academia.edu/LevShchur">Lev Shchur</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Nuclear Physics B - Proceedings Supplements, 1998</p><p class="ds-related-work--abstract ds2-5-body-sm">The probability of simultaneous occurence of at least k spanning clusters has been studied by Monte Carlo simulations on the 2D square lattice at the bond percolation threshold pc = 1/2. The calculated probabilities for free boundary conditions and periodic boundary conditions are in a very good coincidence with the exact formulae developed recently by Cardy.</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 of incipient spanning clusters in critical two-dimensional percolation","attachmentId":43221678,"attachmentType":"pdf","work_url":"https://www.academia.edu/22624017/Probability_of_incipient_spanning_clusters_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-wsj-grid-card-view-pdf" href="https://www.academia.edu/22624017/Probability_of_incipient_spanning_clusters_in_critical_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="7" 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="8" 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="9" data-entity-id="69177707" 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/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-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, 2009</p><p class="ds-related-work--abstract ds2-5-body-sm">It is natural to expect that there are only three possible types of scaling limits for the collection of all percolation interfaces in the plane: (1) a trivial one, consisting of no curves at all, (2) a critical one, in which all points of the plane are surrounded by arbitrarily large loops and every deterministic point is almost surely surrounded</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":"Trivial, Critical and Near-critical Scaling Limits of Two-dimensional Percolation","attachmentId":79372633,"attachmentType":"pdf","work_url":"https://www.academia.edu/69177707/Trivial_Critical_and_Near_critical_Scaling_Limits_of_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/69177707/Trivial_Critical_and_Near_critical_Scaling_Limits_of_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></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":116433199,"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":116433199,"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_116433199" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. 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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-related-work-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><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 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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" href="https://www.academia.edu/117780770/Continuum_Limits_for_Critical_Percolation_and_Other_Stochastic_Geometric_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-related-work-sidebar-card" data-collection-position="6" data-entity-id="69177696" 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/69177696/Cardy_s_formula_for_some_dependent_percolation_models">Cardy’s formula for some dependent percolation models</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">Bulletin of the Brazilian Mathematical 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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" 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href="https://www.academia.edu/48351117/Exact_factorization_of_correlation_functions_in_two_dimensional_critical_percolation">Exact factorization of correlation functions in two-dimensional 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="282899" href="https://umich.academia.edu/RobertZiff">Robert Ziff</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2007</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":"Exact factorization of correlation functions in two-dimensional critical percolation","attachmentId":67009782,"attachmentType":"pdf","work_url":"https://www.academia.edu/48351117/Exact_factorization_of_correlation_functions_in_two_dimensional_critical_percolation","alternativeTracking":true}"><span 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