Optimal Mixing via Tensorization for Random Independent Sets on Arbitrary Trees

TL;DR

This work proves that Glauber dynamics for sampling from the hard-core Gibbs distribution on any $n$-vertex tree mixes in $O(n)$ time for $\lambda<1.1$ (unweighted independent sets), with no dependence on the maximum degree. It introduces an inductive variance-tensorization framework that leverages the tree's combinatorial structure, enabling optimal relaxation without spatial mixing assumptions. By establishing spectral independence for trees up to $\lambda\le 1.3$, it extends rapid mixing results beyond theuniqueness region and connects to broader mixing-time theory via existing spectral-independence results. The combination of variance-factorization techniques and SI-based arguments yields robust, degree-free mixing guarantees and new insights into the reconstruction vs. non-reconstruction regime for trees. The practical impact lies in providing near-optimal, degree-agnostic convergence rates for sampling independent sets on trees, with implications for related combinatorial models on sparse graphs.

Abstract

We study the mixing time of the single-site update Markov chain, known as the Glauber dynamics, for generating a random independent set of a tree. Our focus is obtaining optimal convergence results for arbitrary trees. We consider the more general problem of sampling from the Gibbs distribution in the hard-core model where independent sets are weighted by a parameter $λ>0$; the special case $λ=1$ corresponds to the uniform distribution over all independent sets. Previous work of Martinelli, Sinclair and Weitz (2004) obtained optimal mixing time bounds for the complete $Δ$-regular tree for all $λ$. However, Restrepo et al. (2014) showed that for sufficiently large $λ$ there are bounded-degree trees where optimal mixing does not hold. Recent work of Eppstein and Frishberg (2022) proved a polynomial mixing time bound for the Glauber dynamics for arbitrary trees, and more generally for graphs of bounded tree-width. We establish an optimal bound on the relaxation time (i.e., inverse spectral gap) of $O(n)$ for the Glauber dynamics for unweighted independent sets on arbitrary trees. We stress that our results hold for arbitrary trees and there is no dependence on the maximum degree $Δ$. Interestingly, our results extend (far) beyond the uniqueness threshold which is on the order $λ=O(1/Δ)$. Our proof approach is inspired by recent work on spectral independence. In fact, we prove that spectral independence holds with a constant independent of the maximum degree for any tree, but this does not imply mixing for general trees as the optimal mixing results of Chen, Liu, and Vigoda (2021) only apply for bounded degree graphs. We instead utilize the combinatorial nature of independent sets to directly prove approximate tensorization of variance via a non-trivial inductive proof.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Remark 1.4
• Definition 2.1: Variance Tensorization
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 3.1
• proof : Proof of \ref{['thm:relaxation']}