{"id":357713,"date":"2017-01-25T13:11:31","date_gmt":"2017-01-25T21:11:31","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/?post_type=msr-research-item&p=357713"},"modified":"2018-10-16T20:00:28","modified_gmt":"2018-10-17T03:00:28","slug":"mixing-time-near-critical-random-graphs","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/mixing-time-near-critical-random-graphs\/","title":{"rendered":"Mixing Time Of Near-Critical Random Graphs"},"content":{"rendered":"

Let C<\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> be the largest component of the Erd\\H{o}s–R\\'{e}nyi random graph G<\/span><\/span><\/span>(<\/span>n<\/span>,<\/span>p<\/span>)<\/span><\/span><\/span><\/span>. The mixing time of random walk on C<\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> in the strictly supercritical regime, p<\/span>=<\/span>c<\/span>\/<\/span><\/span><\/span>n<\/span><\/span><\/span><\/span> with fixed c<\/span>><\/span>1<\/span><\/span><\/span><\/span>, was shown to have order log<\/span>2<\/span><\/span><\/span>n<\/span><\/span><\/span><\/span> by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald. In the critical window, p<\/span>=<\/span>(<\/span>1<\/span>+<\/span>\u03b5<\/span>)<\/span>\/<\/span><\/span><\/span>n<\/span><\/span><\/span><\/span> where \u03bb<\/span>=<\/span>\u03b5<\/span>3<\/span><\/span>n<\/span><\/span><\/span><\/span> is bounded, Nachmias and Peres proved that the mixing time on C<\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> is of order n<\/span><\/span><\/span><\/span>. However, it was unclear how to interpolate between these results, and estimate the mixing time as the giant component emerges from the critical window. Indeed, even the asymptotics of the diameter of C<\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> in this regime were only recently obtained by Riordan and Wormald, as well as the present authors and Kim. In this paper, we show that for p<\/span>=<\/span>(<\/span>1<\/span>+<\/span>\u03b5<\/span>)<\/span>\/<\/span><\/span><\/span>n<\/span><\/span><\/span><\/span> with \u03bb<\/span>=<\/span>\u03b5<\/span>3<\/span><\/span>n<\/span>\u2192<\/span>\u221e<\/span><\/span><\/span><\/span> and \u03bb<\/span>=<\/span>o<\/span>(<\/span>n<\/span>)<\/span><\/span><\/span><\/span>, the mixing time on C<\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> is with high probability of order (<\/span>n<\/span>\/<\/span><\/span><\/span>\u03bb<\/span>)<\/span>log<\/span>2<\/span><\/span><\/span>\u03bb<\/span><\/span><\/span><\/span>. In addition, we show that this is the order of the largest mixing time over all components, both in the slightly supercritical and in the slightly subcritical regime [i.e., p<\/span>=<\/span>(<\/span>1<\/span>\u2212<\/span>\u03b5<\/span>)<\/span>\/<\/span><\/span><\/span>n<\/span><\/span><\/span><\/span> with \u03bb<\/span><\/span><\/span><\/span> as above].<\/p>\n","protected":false},"excerpt":{"rendered":"

Let C1 be the largest component of the Erd\\H{o}s–R\\'{e}nyi random graph G(n,p). The mixing time of random walk on C1 in the strictly supercritical regime, p=c\/n with fixed c>1, was shown to have order log2n by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald. In the critical window, p=(1+\u03b5)\/n where \u03bb=\u03b53n is bounded, […]<\/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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