Criticality in random transposition random walk

Időpont: 
2018. March 08. 16:15 to 18:00
Helyszín: 
Building H room 306
Kategória: 
Előadás
Szervezés: 
BME-egyetem
Kapcsolattartó: 
Department of Stochastics

Lecturer: Dominic Yeo (Technion, Haifa)

Abtract: The random walk on the permutations of [N] generated 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 distribution of the sizes of the largest cycles concentrates (after rescaling) on the Poisson-Dirichlet distribution PD(0,1) considerably earlier, after (1+\epsilon)N/2 steps. We show that this behaviour truly 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 classical Erdos-Renyi random graph process, the metric scaling limits of a uniformly-chosen connected graph with a fixed finite number of surplus edges, and analysing the directed cycle structure of large 3-regular graphs. Joint work with Christina Goldschmidt.