Approximate Colorwise Tensorization of Entropy and Optimal Mixing of the Wang-Swendsen-Kotecký Dynamics

2026-07-11

AI Summary

This work addresses the longstanding challenge of analyzing the mixing time of Wang–Swendsen–Kotecký (WSK) dynamics for uniform sampling of proper $q$-colorings. It introduces a novel framework based on relative entropy contraction, pioneering a color-wise entropy decomposition perspective that transcends the limitations of conventional vertex-wise analyses. By developing approximate color-wise tensorization of entropy (ACTE) and approximate color-wise subadditivity of entropy (ACSE) criteria, and integrating them with a local-to-global induction technique from graph theory, the authors establish a new family of functional inequalities tailored to high-dimensional settings. This approach yields optimal $O_q(\log n)$ mixing time bounds for WSK dynamics on chordal and outerplanar graphs, encompassing both vertex and edge coloring of trees and their line graphs. The results not only surpass the irreducibility threshold barriers inherent to Glauber dynamics but also significantly improve upon the best-known mixing time bounds.

Abstract

We study the mixing time of Wang-Swendsen-Kotecký (WSK) dynamics for uniformly sampling proper $q$-colorings. The WSK dynamics is widely used in statistical physics for sampling from the antiferromagnetic Potts model and can be considered a global counterpart of the flip dynamics, which currently yields the state-of-the-art bounds for sampling colorings in general graphs (Carlson and Vigoda, SODA 2025). However, despite its importance, the tools for analyzing such dynamics remain limited. We develop new tools that enable us to analyze the mixing time of the WSK dynamics through the lens of relative entropy contraction. We introduce new criteria for multi-spin distributions: approximate colorwise tensorization of entropy (ACTE) and approximate colorwise subadditivity of entropy (ACSE). These criteria provide a colorwise counterpart to standard vertex-wise entropy factorization principles, and expose a form of color symmetry beyond coordinate-wise analyses. We also develop new inductive approaches for establishing such criteria on specific types of graphs, which can be viewed as local-to-global arguments for proving high-dimensional functional inequalities in a graph-theoretic sense. As concrete applications, we establish an optimal $O_q(\log n)$ mixing time for the WSK dynamics on chordal and outerplanar graphs, down to the optimal number of colors. Because trees and line graphs of trees are chordal, the result covers both vertex and edge colorings of trees. Our results work in a regime that bypasses the irreducibility threshold for Glauber dynamics while also improving the best known mixing time bounds (Carlson, Chen, Feng and Vigoda, SODA 2025).

Problem

• mixing time
• WSK dynamics
• q-colorings
• entropy contraction
• Potts model

Innovation

• approximate colorwise tensorization
• entropy contraction
• WSK dynamics
• mixing time
• functional inequalities