Some circumstances where extra updates can delay mixing

TL;DR

This paper investigates the limits of the censoring inequality in Glauber dynamics, showing it holds for some models like Ising but fails for others like proper colorings and permutations, highlighting boundaries of this property.

Contribution

The paper provides counterexamples demonstrating the failure of the censoring inequality in certain Glauber dynamics models, answering open questions and clarifying its applicability.

Findings

Censoring inequality holds for monotone spin systems like Ising.

Counterexamples show failure in proper graph colorings.

Failure also observed in lazy transpositions on permutations.

Abstract

Peres and Winkler proved a "censoring" inequality for Glauber dynamics on monotone spins systems such as the Ising model. Specifically, if, starting from a constant-spin configuration, the spins are updated at some sequence of sites, then inserting another site into this sequence brings the resulting configuration closer in total variation to the stationary distribution. We show by means of simple counterexamples that the analogous statements fail for Glauber dynamics on proper colorings of a graph, and for lazy transpositions on permutations, answering two questions of Peres. It is not known whether the censoring property holds in other natural settings such as the Potts model.