{"id":186462,"date":"2011-06-30T00:00:00","date_gmt":"2011-07-01T21:58:36","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/the-percolation-phase-transition-in-the-hamming-cube\/"},"modified":"2016-08-22T11:31:18","modified_gmt":"2016-08-22T18:31:18","slug":"the-percolation-phase-transition-in-the-hamming-cube","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/the-percolation-phase-transition-in-the-hamming-cube\/","title":{"rendered":"The percolation phase transition in the Hamming cube"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Consider percolation on the Hamming cube 0,1<sup>n<\/sup> at the critical probability p<sub>c<\/sub> (at which the expected cluster size is 2<sup>n\/3<\/sup>). It is known that if p=p<sub>c<\/sub>(1+O(2<sup>-n\/3<\/sup>), then the largest component is of size roughly 2<sup>2n\/3<\/sup> with high probability and that this quantity is non-concentrated. We show that for any sequence eps(n) such that eps(n)2<sup>-n\/3<\/sup> and eps(n)=o(1) percolation at p<sub>c<\/sub>(1+eps(n)) yields a largest cluster of size (2+o(1))eps(n)2<sup>n<\/sup>.<\/p>\n<p>This result settles a conjecture of Borgs, Chayes, van der Hofstad, Slade and Spencer.<\/p>\n<p>Joint work with Remco van der Hofstad.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Consider percolation on the Hamming cube 0,1n at the critical probability pc (at which the expected cluster size is 2n\/3). It is known that if p=pc(1+O(2-n\/3), then the largest component is of size roughly 22n\/3 with high probability and that this quantity is non-concentrated. We show that for any sequence eps(n) such that eps(n)2-n\/3 and [&hellip;]<\/p>\n","protected":false},"featured_media":196221,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr_hide_image_in_river":0,"footnotes":""},"research-area":[],"msr-video-type":[],"msr-locale":[268875],"msr-post-option":[],"msr-session-type":[],"msr-impact-theme":[],"msr-pillar":[],"msr-episode":[],"msr-research-theme":[],"class_list":["post-186462","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/KukOf8HTLyU","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/186462","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-video"}],"version-history":[{"count":0,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/186462\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/196221"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=186462"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=186462"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=186462"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=186462"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=186462"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=186462"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=186462"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=186462"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=186462"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=186462"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}