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CATEGORIES:Probability
SUMMARY:Criticality in random transposition random walk -
Dominic Yeo (Technion\, Haifa)
DTSTART;TZID=Europe/London:20180227T161500
DTEND;TZID=Europe/London:20180227T171500
UID:TALK97873AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/97873
DESCRIPTION:The random walk on the permutations of [N] generat
ed by the transpositions was shown by Diaconis and
Shahshahani to mix with sharp cutoff around N log
N /2 steps. However\, Schramm showed that the dis
tribution of the sizes of the largest cycles conce
ntrates (after rescaling) on the Poisson-Dirichlet
distribution PD(0\,1) considerably earlier\, afte
r (1+\\epsilon)N/2 steps. We show that this behavi
our in fact emerges precisely during the critical
window of (1+\\lambda N^{-1/3}) N/2 steps\, as \\
lambda \\rightarrow\\infty. Our methods are rather
different\, and involve an analogy with the class
ical Erdos-Renyi random graph process\, the metric
scaling limits of a uniformly-chosen connected gr
aph with a fixed finite number of surplus edges\,
and analysing the directed cycle structure of larg
e 3-regular graphs. Joint work with Christina Gold
schmidt.
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0W
B
CONTACT:Perla Sousi
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