{"id":341732,"date":"2016-12-27T10:47:22","date_gmt":"2016-12-27T18:47:22","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;p=341732"},"modified":"2018-10-16T21:25:40","modified_gmt":"2018-10-17T04:25:40","slug":"slow-convergence-bootstrap-percolation","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/slow-convergence-bootstrap-percolation\/","title":{"rendered":"Slow Convergence in Bootstrap Percolation"},"content":{"rendered":"<p>In the bootstrap percolation model, sites in an <em>L<\/em> by <em>L<\/em> square are initially infected independently with probability <em>p.<\/em> At subsequent steps, a healthy site becomes infected if it has at least 2 infected neighbours. As (<em>L, p<\/em>) \u2192 (\u221e, 0), the probability that the entire square is eventually infected is known to undergo a phase transition in the parameter <em>p<\/em> log <em>L<\/em>, occurring asymptotically at \u03bb = \u03c0<sup>2<\/sup>\/18 [15]. We prove that the discrepancy between the critical parameter and its limit \u03bb is at least \u2126((log <em>L<\/em>) <sup>\u22121\/2<\/sup>). In contrast, the critical window has width only \u0398((log<em> L<\/em>)<sup>\u22121<\/sup>). For the so-called modified model, we prove rigorous explicit bounds which imply for example that the relative discrepancy is at least 1% even when <em>L<\/em> = 10<sup>3000<\/sup>. Our results shed some light on the observed differences between simulations and rigorous asymptotics.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In the bootstrap percolation model, sites in an L by L square are initially infected independently with probability p. At subsequent steps, a healthy site becomes infected if it has at least 2 infected neighbours. As (L, p) \u2192 (\u221e, 0), the probability that the entire square is eventually infected is known to undergo a [&hellip;]<\/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":"","msr-author-ordering":null,"msr_publishername":"Institute of Mathematical Statistics","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"The Annals of Applied 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