Rapid Mixing from Spectral Independence beyond the Boolean Domain

Abstract

We extend the notion of spectral independence (introduced by Anari, Liu, and Oveis Gharan [4]) 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 Δprovided q≥ (α∗ +)Δwhere α∗≈ 1.763 is the unique solution to α∗ = exp (1/α∗) and THORN 0 is any constant. This is the first efficient algorithm for sampling proper q-colourings in this regime with possibly unbounded ". Our main tool of establishing spectral independence is the recursive coupling by Goldberg, Martin, and Paterson [25].