Abstract

We extend the notion of spectral independence (introduced by Anari, Liu, and Oveis Gharan [ALO20]) from the Boolean domain to general discrete domains. This property characterises distributions with limited correlations, and implies that the corresponding Glauber dynamics is rapidly mixing. As a concrete application, we show that Glauber dynamics for sampling proper $q$-colourings mixes in polynomial-time for the family of triangle-free graphs with maximum degree $\Delta$ provided $q\ge (\alpha^*+\delta)\Delta$ where $\alpha^*\approx 1.763$ is the unique solution to $\alpha^*=\exp(1/\alpha^*)$ and $\delta>0$ is any constant. This is the first efficient algorithm for sampling proper $q$-colourings in this regime with possibly unbounded $\Delta$. Our main tool of establishing spectral independence is the recursive coupling by Goldberg, Martin, and Paterson [GMP05].

Highlights

  • Let V be a set of variables, each of which takes values from a discrete domain of size q ≥ 2

  • We will focus on Glauber dynamics in this paper, which is one of the simplest and most widely used Markov chains

  • We show that our notion of spectral independence can be used to obtain efficient sampling algorithms up to known correlation decay regime for multi-spin systems [19, 18]

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Summary

Introduction

Let V be a set of variables, each of which takes values from a discrete domain of size q ≥ 2. In order to apply the result of Alev and Lau [1], they introduced spectral independence, which is the property that the correlation matrix of μ and all of its conditional distributions have bounded maximum eigenvalues. This generalises a similar result by Anari, Liu and Oveis Gharan [2] for q = 2 Their proof is based on a linear algebra argument which completely characterises the spectrum of their influence matrix in terms of the spectrum of the local random walks, so that the result of Alev and Lau [1] applies. We show that our notion of spectral independence can be used to obtain efficient sampling algorithms up to known correlation decay regime for multi-spin systems [19, 18]

Application to spin systems
Preliminaries
Total variation distance and coupling
Markov chain and mixing time Let Ω be a finite set which is the state space
Proof overview
Glauber dynamics and local random walks To prove
Analysis of local random walks
Proof of main theorem It is straightforward to verify that
Simplicial complexes and Glauber dynamics
Simplicial complexes and random walks Let U be a ground set
Connections to Glauber dynamics
Proof of
A coupling based analysis We now prove
Rapid mixing for list colourings
Analysis of mixing time
An easy coupling analysis We now prove the first part of
Influence bounds via recursion Now we prove
Establish recursion via coupling Next, we prove
Tightness of the marginal upper bound
Full Text
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