## Criticality in random transposition random walk

### With Dominic Yeo (Technion, Haifa)

# Criticality in random transposition random walk

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 in fact emerges precisely during the critical window of (1+lambda N^{-1/3}) N/2 steps, as lambda rightarrowinfty. 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.

- Speaker: Dominic Yeo (Technion, Haifa)
- Tuesday 27 February 2018, 16:15–17:15
- Venue: MR12, CMS, Wilberforce Road, Cambridge, CB3 0WB.
- Series: Probability; organiser: Perla Sousi.