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SUMMARY:Criticality in random transposition random walk - Dominic Yeo (Tec
 hnion\, Haifa)
DTSTART:20180227T161500Z
DTEND:20180227T171500Z
UID:TALK97873@talks.cam.ac.uk
CONTACT:Perla Sousi
DESCRIPTION:The random walk on the permutations of [N] generated by the tr
 anspositions was shown by Diaconis and Shahshahani to mix with sharp cutof
 f around N log N /2 steps. However\, Schramm showed that the distribution 
 of the sizes of the largest cycles concentrates (after rescaling) on the P
 oisson-Dirichlet distribution PD(0\,1) considerably earlier\, after (1+\\e
 psilon)N/2 steps. We show that this behaviour in fact emerges precisely du
 ring the critical window of  (1+\\lambda N^{-1/3}) N/2 steps\, as \\lambda
  \\rightarrow\\infty. Our methods are rather different\, and involve an an
 alogy with the classical Erdos-Renyi random graph process\, the metric sca
 ling limits of a uniformly-chosen connected graph with a fixed finite numb
 er of surplus edges\, and analysing the directed cycle structure of large 
 3-regular graphs. Joint work with Christina Goldschmidt.
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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