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class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/122977446/On_the_density_of_2D_critical_percolation_gaskets_and_anchored_clusters"><img alt="Research paper thumbnail of On the density of 2D critical percolation gaskets and anchored clusters" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/122977446/On_the_density_of_2D_critical_percolation_gaskets_and_anchored_clusters">On the density of 2D critical percolation gaskets and anchored clusters</a></div><div class="wp-workCard_item"><span>letters in mathematical physics/Letters in mathematical physics</span><span>, Mar 15, 2024</span></div><div class="wp-workCard_item wp-workCard--actions"><span 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We prove that the Ursell functions u 2k satisfy: (−1) k−1 u 2k is increasing in each interaction. 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The process generates random fractal sets by an iterative procedure which starts by dividing the unit cube [0, 1] d in N d subcubes, and independently retaining or discarding each subcube with probability p or 1 − p respectively. This step is then repeated within the retained subcubes at all scales. As p is varied, there is a percolation phase transition in terms of paths for all d ≥ 2, and in terms of (d − 1)-dimensional \"sheets\" for all d ≥ 3. For any d ≥ 2, we consider the random fractal set produced at the path-percolation critical value p c (N, d), and show that the probability that it contains a path connecting two opposite faces of the cube [0, 1] d tends to one as N → ∞. As an immediate consequence, we obtain that the above probability has a discontinuity, as a function of p, at p c (N, d) for all N sufficiently large. This had previously been proved only for d = 2 (for any N ≥ 2). For d ≥ 3, we prove analogous results for sheet-percolation. 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We review some of the recent progress on the scaling limit of two-dimensional critical ...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">ABSTRACT. We review some of the recent progress on the scaling limit of two-dimensional critical percolation; in particular, the convergence of the exploration path to chordal SLE6 and the full scaling limit of cluster interface loops. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="69177701"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/69177701/Clusters_and_recurrence_in_the_two_dimensional_zero_temperature_stochastic_Ising_model"><img alt="Research paper thumbnail of Clusters and recurrence in the two-dimensional zero-temperature stochastic Ising model" class="work-thumbnail" src="https://attachments.academia-assets.com/79372650/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/69177701/Clusters_and_recurrence_in_the_two_dimensional_zero_temperature_stochastic_Ising_model">Clusters and recurrence in the two-dimensional zero-temperature stochastic Ising model</a></div><div class="wp-workCard_item"><span>The Annals of Applied …</span><span>, 2002</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We analyze clustering and (local) recurrence of a standard Markov process model of spatial domain...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We analyze clustering and (local) recurrence of a standard Markov process model of spatial domain coarsening. The continuous time process, whose state space consists of assignments of $+ 1$ or $-1$ to each site in $\ mathbf {Z}^ 2$, is the zero-temperature limit ...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="758b4dab005c268449b9bfc131f80bf8" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":79372650,"asset_id":69177701,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/79372650/download_file?st=MTczMjc0NTk1MSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="69177701"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="69177701"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 69177701; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=69177701]").text(description); $(".js-view-count[data-work-id=69177701]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 69177701; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='69177701']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 69177701, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "758b4dab005c268449b9bfc131f80bf8" } } $('.js-work-strip[data-work-id=69177701]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":69177701,"title":"Clusters and recurrence in the two-dimensional zero-temperature stochastic Ising model","translated_title":"","metadata":{"abstract":"We analyze clustering and (local) recurrence of a standard Markov process model of spatial domain coarsening. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="69177700"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/69177700/The_scaling_limit_geometry_of_near_critical_2D_percolation"><img alt="Research paper thumbnail of The scaling limit geometry of near-critical 2D percolation" class="work-thumbnail" src="https://attachments.academia-assets.com/79372641/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" 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><div class="wp-workCard_item"><span>Journal of statistical physics</span><span>, 2006</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We analyze the geometry of scaling limits of near-critical 2D percolation, ie, for p= p c+ λδ 1/ν...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">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 ...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="6b075f6ca6d5d0600d7b50c8b1368d27" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":79372641,"asset_id":69177700,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/79372641/download_file?st=MTczMjc0NTk1MSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="69177700"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="69177700"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 69177700; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=69177700]").text(description); $(".js-view-count[data-work-id=69177700]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 69177700; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='69177700']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 69177700, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "6b075f6ca6d5d0600d7b50c8b1368d27" } } $('.js-work-strip[data-work-id=69177700]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":69177700,"title":"The scaling limit geometry of near-critical 2D percolation","translated_title":"","metadata":{"abstract":"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. 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It combines the continuum nonsimple loop ...","internal_url":"https://www.academia.edu/69177700/The_scaling_limit_geometry_of_near_critical_2D_percolation","translated_internal_url":"","created_at":"2022-01-22T09:14:23.989-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":52848710,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":79372641,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/79372641/thumbnails/1.jpg","file_name":"0510740.pdf","download_url":"https://www.academia.edu/attachments/79372641/download_file?st=MTczMjc0NTk1MSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_scaling_limit_geometry_of_near_criti.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/79372641/0510740-libre.pdf?1642873946=\u0026response-content-disposition=attachment%3B+filename%3DThe_scaling_limit_geometry_of_near_criti.pdf\u0026Expires=1732749551\u0026Signature=FxQQdWdkR0phgRUxOZBVMdgkGpI9aFq3pCtYSBz3wGL-64hNQa~L0gnLdgRqC4hlD3qHxJxrcQnhWzmdFLEDnz-9rJ2e3C3rjJ4BAD-efR1wUlitSob9T~wTY~jO665aeEpyj-wnzrlB6khF~gV78L~A83ToK1ArRfJX24WPSePbGw~qQrBSwGEvuAEE97rXyBzi1zgT90DxjUvpEDXpzld2JuE4c0vEk~CUhk6qrWn7JuG82ahksNM0YlWKNiY3fUtF5RpkSgq3nl6-hkuRPOnmDMzRYu6-9auqnl00T3C5kK2Z7kmIpGz0v93Blvb5GgAKv0Q62-0N0-ufcp4xnQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"The_scaling_limit_geometry_of_near_critical_2D_percolation","translated_slug":"","page_count":21,"language":"en","content_type":"Work","owner":{"id":52848710,"first_name":"Federico","middle_initials":null,"last_name":"Camia","page_name":"FedericoCamia","domain_name":"independent","created_at":"2016-09-01T22:54:59.634-07:00","display_name":"Federico Camia","url":"https://independent.academia.edu/FedericoCamia"},"attachments":[{"id":79372641,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/79372641/thumbnails/1.jpg","file_name":"0510740.pdf","download_url":"https://www.academia.edu/attachments/79372641/download_file?st=MTczMjc0NTk1MSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_scaling_limit_geometry_of_near_criti.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/79372641/0510740-libre.pdf?1642873946=\u0026response-content-disposition=attachment%3B+filename%3DThe_scaling_limit_geometry_of_near_criti.pdf\u0026Expires=1732749551\u0026Signature=FxQQdWdkR0phgRUxOZBVMdgkGpI9aFq3pCtYSBz3wGL-64hNQa~L0gnLdgRqC4hlD3qHxJxrcQnhWzmdFLEDnz-9rJ2e3C3rjJ4BAD-efR1wUlitSob9T~wTY~jO665aeEpyj-wnzrlB6khF~gV78L~A83ToK1ArRfJX24WPSePbGw~qQrBSwGEvuAEE97rXyBzi1zgT90DxjUvpEDXpzld2JuE4c0vEk~CUhk6qrWn7JuG82ahksNM0YlWKNiY3fUtF5RpkSgq3nl6-hkuRPOnmDMzRYu6-9auqnl00T3C5kK2Z7kmIpGz0v93Blvb5GgAKv0Q62-0N0-ufcp4xnQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":16460,"name":"Statistical Physics","url":"https://www.academia.edu/Documents/in/Statistical_Physics"},{"id":33069,"name":"Probability","url":"https://www.academia.edu/Documents/in/Probability"},{"id":50071,"name":"Percolation","url":"https://www.academia.edu/Documents/in/Percolation"},{"id":80414,"name":"Mathematical Sciences","url":"https://www.academia.edu/Documents/in/Mathematical_Sciences"},{"id":118582,"name":"Physical sciences","url":"https://www.academia.edu/Documents/in/Physical_sciences"},{"id":310813,"name":"Criticality","url":"https://www.academia.edu/Documents/in/Criticality"},{"id":595394,"name":"Continuum","url":"https://www.academia.edu/Documents/in/Continuum"},{"id":1155889,"name":"Minimal Spanning Tree","url":"https://www.academia.edu/Documents/in/Minimal_Spanning_Tree"},{"id":1317328,"name":"Scaling Limits","url":"https://www.academia.edu/Documents/in/Scaling_Limits"},{"id":1495455,"name":"Finite Size Scaling","url":"https://www.academia.edu/Documents/in/Finite_Size_Scaling"},{"id":2248009,"name":"Continuo","url":"https://www.academia.edu/Documents/in/Continuo"}],"urls":[{"id":16795167,"url":"http://direct.bl.uk/research/58/54/RN200090212.html"}]}, dispatcherData: dispatcherData }); 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For a particular choice of loop soup intensity, we show that the corresponding spin system is reflection-positive and is dual, in the Kramers-Wannier sense, to the spin system sgn(ϕ) where ϕ is a discrete Gaussian free field. In general, we show that the spin correlation functions have conformally covariant scaling limits corresponding to the one-parameter family of functions studied by Camia, Gandolfi and Kleban (Nuclear Physics B 902, 2016) and defined in terms of the winding of the Brownian loop soup. These functions have properties consistent with the behavior of correlation functions of conformal primaries in a conformal field theory. 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We prove that the Ursell functions u 2k satisfy: (−1) k−1 u 2k is increasing in each interaction. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="110282453"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/110282453/Conformal_Measure_Ensembles_for_Percolation_and_the_FK_Ising_model"><img alt="Research paper thumbnail of Conformal Measure Ensembles for Percolation and the FK-Ising model" class="work-thumbnail" src="https://attachments.academia-assets.com/108147751/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110282453/Conformal_Measure_Ensembles_for_Percolation_and_the_FK_Ising_model">Conformal Measure Ensembles for Percolation and the FK-Ising model</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Jul 6, 2015</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="8b1a87ac7adb7221fa279f6f09bca999" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108147751,"asset_id":110282453,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108147751/download_file?st=MTczMjc0NTk1Miw4LjIyMi4yMDguMTQ2&st=MTczMjc0NTk1MSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110282453"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110282453"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110282453; 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Our results apply, in particular, to critical Bernoulli site percolation on the triangular lattice and to the critical FK-Ising model on the square lattice. Fundamental properties of the scaling limit, such as conformal covariance, are explored. As an application to Bernoulli percolation, we obtain the scaling limit of the largest cluster in a bounded domain. 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We conjecture that the massive Brownian loop soup describes the zero level lines of the massive Gaussian free field, and discuss possible relations to other models, such as Ising, in the offcritical regime.","publication_date":{"day":null,"month":null,"year":2016,"errors":{}},"grobid_abstract_attachment_id":105928964},"translated_abstract":null,"internal_url":"https://www.academia.edu/107170212/Massive_Brownian_Loop_Soup","translated_internal_url":"","created_at":"2023-09-25T06:14:50.437-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":52848710,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":105928964,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/105928964/thumbnails/1.jpg","file_name":"1309.6068.pdf","download_url":"https://www.academia.edu/attachments/105928964/download_file?st=MTczMjc0NTk1Miw4LjIyMi4yMDguMTQ2&st=MTczMjc0NTk1MSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Massive_Brownian_Loop_Soup.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/105928964/1309.6068-libre.pdf?1695655883=\u0026response-content-disposition=attachment%3B+filename%3DMassive_Brownian_Loop_Soup.pdf\u0026Expires=1732749551\u0026Signature=Wo8SiNJzaxh9vykA71nQq3Oh1Suv63i9xKilbMA1QtYm9NVP9-0LzQxRVF9ZlZdBhVtYhH3EV2EOLGI97hDSUCbSxnRJLBqXsdpjhyoik0~vs2TnqjoI27iWGsDHzLgxsxg43SUIKn6rQ0qhCMaBS9gD6ZW6UDpa3hGil91ickSz-qOkGKl4cSluqQ2YfVFCmUr8jmXcTEEZ5-xDS0jvo8PIxfasblDqQywPk~2MOAKTN37wgGIT5IPYFvh5Iunh0pslj5URdhuwYIV0mQ0MEe6C1JOE7GNqDFMPUgK-XTYkdtrxzvrpV12yq~dVX0FX8g9OzCIsiKCkT30eNOu6Iw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Massive_Brownian_Loop_Soup","translated_slug":"","page_count":23,"language":"en","content_type":"Work","owner":{"id":52848710,"first_name":"Federico","middle_initials":null,"last_name":"Camia","page_name":"FedericoCamia","domain_name":"independent","created_at":"2016-09-01T22:54:59.634-07:00","display_name":"Federico Camia","url":"https://independent.academia.edu/FedericoCamia"},"attachments":[{"id":105928964,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/105928964/thumbnails/1.jpg","file_name":"1309.6068.pdf","download_url":"https://www.academia.edu/attachments/105928964/download_file?st=MTczMjc0NTk1Miw4LjIyMi4yMDguMTQ2&st=MTczMjc0NTk1MSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Massive_Brownian_Loop_Soup.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/105928964/1309.6068-libre.pdf?1695655883=\u0026response-content-disposition=attachment%3B+filename%3DMassive_Brownian_Loop_Soup.pdf\u0026Expires=1732749551\u0026Signature=Wo8SiNJzaxh9vykA71nQq3Oh1Suv63i9xKilbMA1QtYm9NVP9-0LzQxRVF9ZlZdBhVtYhH3EV2EOLGI97hDSUCbSxnRJLBqXsdpjhyoik0~vs2TnqjoI27iWGsDHzLgxsxg43SUIKn6rQ0qhCMaBS9gD6ZW6UDpa3hGil91ickSz-qOkGKl4cSluqQ2YfVFCmUr8jmXcTEEZ5-xDS0jvo8PIxfasblDqQywPk~2MOAKTN37wgGIT5IPYFvh5Iunh0pslj5URdhuwYIV0mQ0MEe6C1JOE7GNqDFMPUgK-XTYkdtrxzvrpV12yq~dVX0FX8g9OzCIsiKCkT30eNOu6Iw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":192783,"name":"Field","url":"https://www.academia.edu/Documents/in/Field"}],"urls":[{"id":34057693,"url":"http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.756.3768\u0026rep=rep1\u0026type=pdf"}]}, dispatcherData: dispatcherData }); 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The process generates random fractal sets by an iterative procedure which starts by dividing the unit cube [0, 1] d in N d subcubes, and independently retaining or discarding each subcube with probability p or 1 − p respectively. This step is then repeated within the retained subcubes at all scales. As p is varied, there is a percolation phase transition in terms of paths for all d ≥ 2, and in terms of (d − 1)-dimensional \"sheets\" for all d ≥ 3. For any d ≥ 2, we consider the random fractal set produced at the path-percolation critical value p c (N, d), and show that the probability that it contains a path connecting two opposite faces of the cube [0, 1] d tends to one as N → ∞. As an immediate consequence, we obtain that the above probability has a discontinuity, as a function of p, at p c (N, d) for all N sufficiently large. This had previously been proved only for d = 2 (for any N ≥ 2). For d ≥ 3, we prove analogous results for sheet-percolation. 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We review some of the recent progress on the scaling limit of two-dimensional critical ...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">ABSTRACT. We review some of the recent progress on the scaling limit of two-dimensional critical percolation; in particular, the convergence of the exploration path to chordal SLE6 and the full scaling limit of cluster interface loops. 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The continuous time process, whose state space consists of assignments of $+ 1$ or $-1$ to each site in $\ mathbf {Z}^ 2$, is the zero-temperature limit ...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="758b4dab005c268449b9bfc131f80bf8" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":79372650,"asset_id":69177701,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/79372650/download_file?st=MTczMjc0NTk1Miw4LjIyMi4yMDguMTQ2&st=MTczMjc0NTk1MSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="69177701"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="69177701"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 69177701; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=69177701]").text(description); $(".js-view-count[data-work-id=69177701]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 69177701; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='69177701']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 69177701, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "758b4dab005c268449b9bfc131f80bf8" } } $('.js-work-strip[data-work-id=69177701]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":69177701,"title":"Clusters and recurrence in the two-dimensional zero-temperature stochastic Ising model","translated_title":"","metadata":{"abstract":"We analyze clustering and (local) recurrence of a standard Markov process model of spatial domain coarsening. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="69177700"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/69177700/The_scaling_limit_geometry_of_near_critical_2D_percolation"><img alt="Research paper thumbnail of The scaling limit geometry of near-critical 2D percolation" class="work-thumbnail" src="https://attachments.academia-assets.com/79372641/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" 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><div class="wp-workCard_item"><span>Journal of statistical physics</span><span>, 2006</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We analyze the geometry of scaling limits of near-critical 2D percolation, ie, for p= p c+ λδ 1/ν...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">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 ...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="6b075f6ca6d5d0600d7b50c8b1368d27" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":79372641,"asset_id":69177700,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/79372641/download_file?st=MTczMjc0NTk1Miw4LjIyMi4yMDguMTQ2&st=MTczMjc0NTk1MSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="69177700"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="69177700"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 69177700; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=69177700]").text(description); $(".js-view-count[data-work-id=69177700]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 69177700; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='69177700']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 69177700, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "6b075f6ca6d5d0600d7b50c8b1368d27" } } $('.js-work-strip[data-work-id=69177700]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":69177700,"title":"The scaling limit geometry of near-critical 2D percolation","translated_title":"","metadata":{"abstract":"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 ...","publisher":"Springer","publication_date":{"day":null,"month":null,"year":2006,"errors":{}},"publication_name":"Journal of statistical physics"},"translated_abstract":"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. 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