Cutoff for conjugacy-invariant random walks on the permutation group

TL;DR

This paper proves a conjecture about the mixing time of conjugacy-invariant random walks on permutation groups, revealing a phase transition in Ricci curvature related to the walk's convergence rate.

Contribution

It establishes the occurrence of a phase transition in the Ricci curvature for conjugacy-invariant random walks, connecting mixing times with geometric properties.

Findings

Phase transition in Ricci curvature at critical time

Curvature is zero for c ≤ 1, positive for c > 1

Connection between coalescence processes and mixing times

Abstract

We prove a conjecture raised by the work of Diaconis and Shahshahani (1981) about the mixing time of random walks on the permutation group induced by a given conjugacy class. To do this we exploit a connection with coalescence and fragmentation processes and control the Kantorovitch distance by using a variant of a coupling due to Oded Schramm. Recasting our proof in the language of Ricci curvature, our proof establishes the occurrence of a phase transition, which takes the following form in the case of random transpositions: at time $cn/2$, the curvature is asymptotically zero for $c\le 1$ and is strictly positive for $c>1$.