Rapid Mixing from Spectral Independence beyond the Boolean Domain

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 Δ provided 𝑞 ≥ (𝛼 ∗ + 𝛿 )Δ where 𝛼 ∗ ≈ 1.763 is the unique solution to 𝛼 ∗ = exp (1/𝛼 ∗ ) and 𝛿 > 0 is any constant. This is the first efficient algorithm for sampling proper 𝑞-colourings in this regime with possibly unbounded Δ. Our main tool of establishing spectral independence is the recursive coupling by Goldberg, Martin, and Paterson.