Rapid Mixing for Colorings via Spectral Independence

TL;DR

This paper extends the spectral independence method to colorings, proving polynomial mixing times for Glauber dynamics on triangle-free graphs with certain q, improving previous bounds and simplifying analysis.

Contribution

It develops a spectral independence framework for colorings, achieving polynomial mixing times with bounds independent of maximum degree and number of colors, simplifying previous complex methods.

Findings

Polynomial mixing time for q-colorings on triangle-free graphs with q ≥ αΔ+1.

Improved bounds compared to previous methods requiring larger girth or dependence on Δ and q.

Simplified analysis avoiding complex coupling and complex plane techniques.

Abstract

The spectral independence approach of Anari et al. (2020) utilized recent results on high-dimensional expanders of Alev and Lau (2020) and established rapid mixing of the Glauber dynamics for the hard-core model defined on weighted independent sets. We develop the spectral independence approach for colorings, and obtain new algorithmic results for the corresponding counting/sampling problems. Let $\alpha^*\approx 1.763$ denote the solution to $\exp(1/x)=x$ and let $\alpha>\alpha^*$. We prove that, for any triangle-free graph $G=(V,E)$ with maximum degree $\Delta$, for all $q\geq\alpha\Delta+1$, the mixing time of the Glauber dynamics for $q$-colorings is polynomial in $n=|V|$, with the exponent of the polynomial independent of $\Delta$ and $q$. In comparison, previous approximate counting results for colorings held for a similar range of $q$ (asymptotically in $\Delta$) but with larger girth requirement or with a running time where the polynomial exponent depended on $\Delta$ and $q$ (exponentially). One further feature of using the spectral independence approach to study colorings is that it avoids many of the technical complications in previous approaches caused by coupling arguments or by passing to the complex plane; the key improvement on the running time is based on relatively simple combinatorial arguments which are then translated into spectral bounds.