Limit Profile for Projections of Random Walks on Groups

TL;DR

This paper investigates the precise shape of convergence to equilibrium, known as the cutoff profile, for projections of random walks on groups, extending classical methods and analyzing specific processes like the $k$-particle interchange process.

Contribution

It develops new techniques to determine the cutoff profile for projections of conjugacy-invariant random walks on groups, including the $k$-particle interchange process.

Findings

Established cutoff profile for the $k$-particle interchange process.

Extended analysis from conjugacy-invariant walks to certain projections.

Provided a framework for precise convergence shape in Markov chain mixing.

Abstract

Establishing cutoff, an abrupt transition from "not mixed" to "well mixed", is a classical topic in the theory of mixing times for Markov chains. Interest has grown recently in determining not only the existence of cutoff and the order of its mixing time and window, but the exact shape, or profile, of the convergence inside the window. Classical techniques, such as coupling or $\ell_2$-bounds, are typically too crude to establish this and there has been a push to develop general techniques. We build upon this work, extending from conjugacy-invariant random walks on groups to certain projections. We exemplify our method by analysing the $k$-particle interchange process on the complete $n$-graph with $k \asymp n$. This is a projection of the random-transposition card shuffle, which corresponds to $k = n$, analysed by Teyssier.