Convergence rates of Markov chains on spaces of partitions

Harry Crane,Steven Lalley

doi:10.1214/ejp.v18-2389

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.

Highlights

Exchangeability does not imply that the Markov chain has the Feller property, but if a Markov chain is both exchangeable and Feller it has a simple paintbox representation, as proved by Crane [3]
We shall refer to such Markov chains Xt and Xt[n] as exchangeable Feller cut-and-paste chains, or EFCP chains for short
Choose i.i.d. stochastic matrices S1, S2, . . . with law Σ, all independent of X0; second, given X0, S1, S2, . . ., let M1, M2, . . . be conditionally independent k−ary partition matrices with laws Mi ∼ μSi for each i = 1, 2, . . ., and define the cut-andpaste chain Xm by equation (8). This construction is fundamental to our arguments, and so when considering an EFCP chain with directing measure μΣ, we shall assume that it is defined on a probability space together with a paintbox sequence S1, S2, . . . . For each m ∈ N, set

Summary

INTRODUCTION

Markov chain {Xt}t=0,1,2,... on the space [k]N of k−colorings of the positive integers N is said to be exchangeable if its transition law is equivariant with respect to finite permutations of N (that is, permutations that fix all but finitely many elements of N). (see Proposition 3.3 in section 3.3), conditional on the paintbox sequence, the coordinate sequences {Xmi +1}m≥0 are independent, time-inhomogeneous Markov chains on the state space [k] with one-step transition probability matrices S1, S2, . Under mild hypotheses on the paintbox distribution (see the discussion in section 5) the restrictions of EFCP chains Xt[n] to the finite configuration spaces [k][n] are ergodic. In the special case k = 2 the results are related to some classical results for random walks on the hypercube, e.g. the Ehrenfest chain on {0, 1}n: see example 5.9 The key to both results is that the relative frequencies of the different colors are determined by the random matrix products StSt−1 · · · S1 (see Proposition 3.3).

PRELIMINARIES

RANDOM STOCHASTIC MATRIX PRODUCTS

CONVERGENCE TO STATIONARITY OF EFCP CHAINS

PROJECTED CUT-AND-PASTE CHAINS