We address the question of \fnite-size scaling in percolation by studying bond percolation in a \fnite box of side length n, both in two and in higher dimensions. In dimension d = 2, we obtain a complete characterization of \fnite-size scaling. In dimensions d > 2, we establish the same results under a set of hypotheses related to so-called scaling and hyperscaling postulates which are widely believed to hold up to d = 6. As a function of the size of the box, we determine the scaling window in which the system behaves critically. We characterize criticality in terms of the scaling of the sizes of the largest clusters in the box: incipient in\fnite clusters which give rise to the in\fnite cluster. Within the scaling window, we show that the size of the largest cluster behaves like nd\u0019n, where \u0019n is the probability at criticality that the origin is connected to the boundary of a box of radius n. We also show that, inside the window, there are typically many clusters of scale nd\u0019n, and hence that \\the” incipient in\fnite cluster is not unique. Below the window, we show that the size of the largest cluster scales like \u0018d\u0019\u0018 log(n=\u0018), where \u0018 is the correlation length, and again, there are many clusters of this scale. Above the window, we show that the size of the largest cluster scales like ndP1, where P1 is the in\fnite cluster density, and that there is only one cluster of this scale. Our results are \fnite-dimensional analogues of results on the dominant component of the Erd}os-R\u0013enyi mean-fi\feld random graph model.<\/p>\n","protected":false},"excerpt":{"rendered":"
We address the question of \fnite-size scaling in percolation by studying bond percolation in a \fnite box of side length n, both in two and in higher dimensions. In dimension d = 2, we obtain a complete characterization of \fnite-size scaling. In dimensions d > 2, we establish the same results under a set of […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13561],"msr-publication-type":[193716],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-317690","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-locale-en_us"],"msr_publishername":"","msr_edition":"Communications in Mathematical Physics 224","msr_affiliation":"","msr_published_date":"2000-12-01","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"153-204","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"459564","msr_publicationurl":"","msr_doi":"","msr_publication_uploader":[{"type":"file","title":"the-birth-of-the-infinite-cluster-finite-size-scaling-in-percolation","viewUrl":"https:\/\/www.microsoft.com\/en-us\/research\/uploads\/prod\/2016\/11\/The-Birth-of-the-Infinite-Cluster-Finite-size-Scaling-in-Percolation.pdf","id":459564,"label_id":0}],"msr_related_uploader":"","msr_attachments":[],"msr-author-ordering":[{"type":"user_nicename","value":"borgs","user_id":31278,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=borgs"},{"type":"user_nicename","value":"jchayes","user_id":32200,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=jchayes"},{"type":"text","value":"H. Kesten","user_id":0,"rest_url":false},{"type":"text","value":"J. Spencer","user_id":0,"rest_url":false}],"msr_impact_theme":[],"msr_research_lab":[],"msr_event":[],"msr_group":[],"msr_project":[],"publication":[],"video":[],"download":[],"msr_publication_type":"inproceedings","related_content":[],"_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/317690"}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-research-item"}],"version-history":[{"count":1,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/317690\/revisions"}],"predecessor-version":[{"id":523713,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/317690\/revisions\/523713"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=317690"}],"wp:term":[{"taxonomy":"msr-content-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-content-type?post=317690"},{"taxonomy":"msr-research-highlight","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-highlight?post=317690"},{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=317690"},{"taxonomy":"msr-publication-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-publication-type?post=317690"},{"taxonomy":"msr-product-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-product-type?post=317690"},{"taxonomy":"msr-focus-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-focus-area?post=317690"},{"taxonomy":"msr-platform","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-platform?post=317690"},{"taxonomy":"msr-download-source","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-download-source?post=317690"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=317690"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=317690"},{"taxonomy":"msr-field-of-study","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-field-of-study?post=317690"},{"taxonomy":"msr-conference","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-conference?post=317690"},{"taxonomy":"msr-journal","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-journal?post=317690"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=317690"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=317690"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}