Rapid mixing from spectral independence beyond the Boolean domain

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 $q\ge ({\alpha }^{\ast }+\delta )\Delta$ where ${\alpha }^{\ast }\approx 1.763$ is the unique solution to ${\alpha }^{\ast }=\exp (1/{\alpha }^{\ast })$ 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].