{"id":339968,"date":"2016-12-21T11:14:31","date_gmt":"2016-12-21T19:14:31","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/?post_type=msr-research-item&#038;p=339968"},"modified":"2018-10-16T21:19:03","modified_gmt":"2018-10-17T04:19:03","slug":"pattern-theorem-random-sorting-networks","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/pattern-theorem-random-sorting-networks\/","title":{"rendered":"A Pattern Theorem for Random Sorting Networks"},"content":{"rendered":"<p>A sorting network is a shortest path from 12 \u00b7 \u00b7 \u00b7 <em>n<\/em> to <em>n<\/em> \u00b7 \u00b7 \u00b7 21 in the Cayley graph of the symmetric group <em>S<\/em><sub>n<\/sub> generated by nearest-neighbor swaps. A pattern is a sequence of swaps that forms an initial segment of some sorting network. We prove that in a uniformly random <em>n<\/em>-element sorting network, any fixed pattern occurs in at least <em>cn<\/em><sup>2<\/sup> disjoint space-time locations, with probability tending to 1 exponentially fast as <em>n<\/em> \u2192 \u221e. Here <em>c<\/em> is a positive constant which depends on the choice of pattern. As a consequence, the probability that the uniformly random sorting network is geometrically realizable tends to 0.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A sorting network is a shortest path from 12 \u00b7 \u00b7 \u00b7 n to n \u00b7 \u00b7 \u00b7 21 in the Cayley graph of the symmetric group Sn generated by nearest-neighbor swaps. A pattern is a sequence of swaps that forms an initial segment of some sorting network. We prove that in a uniformly random [&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":"Electronic Journal of 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