Sampling independent sets in the discrete torus

TL;DR

This paper investigates the mixing time of Glauber dynamics for the hard-core model on high-dimensional discrete tori, showing slow convergence for certain activity levels and revealing phase coexistence phenomena.

Contribution

It extends previous results by establishing exponential slow mixing for a broader range of activity parameters and introduces combinatorial enumeration methods to analyze phase coexistence.

Findings

Glauber dynamics mixes slowly for activity x > cd^{-1/4}log^{3/4}d in high dimensions.

The probability of balanced independent sets (equal vertices in bipart partition) is exponentially small.

Results imply phase coexistence and large fluctuations in independent set sizes.

Abstract

The even discrete torus is the graph T_{L,d} on vertex set {0,...,L-1}^d (L even) with two vertices adjacent if they differ by 1 (mod L) on one coordinate. The hard-core measure with activity x on T_{L,d} is the distribution pi_x on the independent sets (sets of vertices spanning no edges) of T_{L,d} in which a set I is chosen with probability proportional to x^|I|. This distribution occurs in problems from statistical physics and communication networks. We study Glauber dynamics, a single-site update Markov chain on the set of independent sets of T_{L,d} whose stationary distribution is pi_x. We show that for x > cd^{-1/4}log^{3/4}d (and d large) the convergence to stationarity is exponentially slow in L^{d-1}. This improves a result of Borgs et al., who had shown slow mixing for x > c^d. Our proof, which extends to r-local chains (chains which alter the state of at most a proportion r of the vertices in each step) for suitable r, follows the conductance argument of Borgs et al., adding to it some combinatorial enumeration methods that are modifications of those used by Galvin and Kahn to show that the hard-core model with parameter x on the integer lattice Z^d exhibits phase coexistence for x > cd^{-1/4}log^{3/4}d. The graph T_{L,d} is bipartite, with partition classes E (the vertices the sum of whose coordinates is even) and O. Our result can be expressed combinatorially as the statement that for each sufficiently large x, there is an r(x)>0 such that if I is an independent set chosen according to pi_x, then the probability that ||I \cap E|-|I \cap O|| is at most r(x)L^d is exponentially small in L^{d-1}. In particular, for all eps>0 the probability that a uniformly chosen independent set from T_{L,d} satisfies ||I \cap E|-|I \cap O|| \leq (.25 - eps)L^d is exponentially small in L^{d-1}.