Convergence Rates of Markov Chains on Spaces of Partitions

TL;DR

This paper investigates the convergence rates of exchangeable partition-valued Markov chains, specifically cut-and-paste chains, establishing upper bounds for mixing times and conditions for the cutoff phenomenon.

Contribution

It introduces a novel analysis of cut-and-paste chains using products of i.i.d. matrices, providing new bounds and conditions for rapid convergence.

Findings

Upper bounds for mixing times of ergodic chains

Conditions under which the cutoff phenomenon occurs

Representation of chain transitions via stochastic matrices

Abstract

We study the convergence rate to stationarity for a class of exchangeable partition-valued Markov chains called cut-and-paste chains. The law governing the transitions of a cut-and-paste chain are determined by products of i.i.d. stochastic matrices, which describe the chain induced on the simplex by taking asymptotic frequencies. Using this representation, we establish upper bounds for the mixing times of ergodic cut-and-paste chains, and under certain conditions on the distribution of the governing random matrices we show that the "cutoff phenomenon" holds.