Strong spatial mixing for colorings on trees and its algorithmic applications
Zongchen Chen, Kuikui Liu, Nitya Mani, Ankur Moitra
TL;DR
This work advances the understanding of correlation decay and efficient sampling for colorings by proving strong spatial mixing on trees of maximum degree $\\Delta$ whenever $q \\ge \\\Delta+3$, significantly improving prior bounds. It introduces a general reduction from large-girth graphs to SSM on trees via a new local coupling, enabling spectral independence and near-optimal Glauber dynamics mixing on graphs with girth $g = \\Omega_\\Delta(1)$ for $q \\ge \\\Delta+3$, with explicit $O(n\log n)$ time guarantees. The core method combines a univariate potential function and a weighted $L^2$-type contraction of the tree recursion to derive SSM, WSM, and spectral independence, and it extends to the antiferromagnetic Potts model to obtain near-tight WSM bounds on trees. The results yield near-optimal algorithmic performance for sampling colorings on broad graph families and illuminate the precise mechanisms by which tree-like structures control global correlation decay. Collectively, they mark a substantial step toward resolving the conjectured thresholds for coloring1 problems and demonstrate a versatile framework connecting contraction on trees to rapid mixing on general graphs.
Abstract
Strong spatial mixing (SSM) is an important quantitative notion of correlation decay for Gibbs distributions arising in statistical physics, probability theory, and theoretical computer science. A longstanding conjecture is that the uniform distribution on proper $q$-colorings on a $Δ$-regular tree exhibits SSM whenever $q \ge Δ+1$. Moreover, it is widely believed that as long as SSM holds on bounded-degree trees with $q$ colors, one would obtain an efficient sampler for $q$-colorings on all bounded-degree graphs via simple Markov chain algorithms. It is surprising that such a basic question is still open, even on trees, but then again it also highlights how much we still have to learn about random colorings. In this paper, we show the following: (1) For any $Δ\ge 3$, SSM holds for random $q$-colorings on trees of maximum degree $Δ$ whenever $q \ge Δ+ 3$. Thus we almost fully resolve the aforementioned conjecture. Our result substantially improves upon the previously best bound which requires $q \ge 1.59Δ+γ^*$ for an absolute constant $γ^* > 0$. (2) For any $Δ\ge 3$ and girth $g = Ω_Δ(1)$, we establish optimal mixing of the Glauber dynamics for $q$-colorings on graphs of maximum degree $Δ$ and girth $g$ whenever $q \ge Δ+3$. Our approach is based on a new general reduction from spectral independence on large-girth graphs to SSM on trees that is of independent interest. Using the same techniques, we also prove near-optimal bounds on weak spatial mixing (WSM), a closely-related notion to SSM, for the antiferromagnetic Potts model on trees.