{"id":340556,"date":"2016-12-22T09:29:23","date_gmt":"2016-12-22T17:29:23","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&#038;p=340556"},"modified":"2018-10-16T21:23:28","modified_gmt":"2018-10-17T04:23:28","slug":"oriented-swap-process","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/oriented-swap-process\/","title":{"rendered":"The Oriented Swap Process"},"content":{"rendered":"<p>Particles labelled 1 \u00b7 \u00b7 \u00b7 <em>n<\/em> are arranged initially in increasing order. Subsequently, each pair of neighbouring particles that is currently in increasing order swaps according to a Poisson process of rate 1. We analyze the asymptotic behaviour of this process as <em>n<\/em> \u2192 \u221e. We prove that the space-time trajectories of individual particles converge (when suitably scaled) to a certain family of random curves with two points of non-differentiability, and that the permutation matrix at a given time converges to a certain deterministic measure with absolutely continuous and singular parts. The absorbing state (where all particles are in decreasing order) is reached at time (2+<em>o<\/em>(1))<em>n<\/em>. The finishing times of individual particles converge to deterministic limits with fluctuations asymptotically governed by the Tracy-Widom distribution.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Particles labelled 1 \u00b7 \u00b7 \u00b7 n are arranged initially in increasing order. Subsequently, each pair of neighbouring particles that is currently in increasing order swaps according to a Poisson process of rate 1. We analyze the asymptotic behaviour of this process as n \u2192 \u221e. We prove that the space-time trajectories of individual particles [&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 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