Combinatorics

Discrete mathematics

Physics

Mathematics

Markov chain

Optics

Statistics

Mathematical analysis

Acoustics

Degree (music)

Scattering

Glauber

Bounded function

We prove an optimal mixing time bound for the single-site update Markov chain known as the Glauber dynamics or Gibbs sampling in a variety of settings. Our work presents an improved version of the spectral independence approach of Anari et al. (2020) and shows O(nlogn) mixing time on any n-vertex graph of bounded degree when the maximum eigenvalue of an associated influence matrix is bounded. As an application of our results, for the hard-core model on independent sets weighted by a fugacity λ, we establish O(nlogn) mixing time for the Glauber dynamics on any n-vertex graph of constant maximum degree Δ when λ<λc(Δ) where λc(Δ) is the critical point for the uniqueness/non-uniqueness phase transition on the Δ-regular tree. More generally, for any antiferromagnetic 2-spin system we prove O(nlogn) mixing time of the Glauber dynamics on any bounded degree graph in the corresponding tree uniqueness region. Our results apply more broadly; for example, we also obtain O(nlogn) mixing for q-colorings of triangle-free graphs of maximum degree Δ when the number of colors satisfies q > α Δ where α ≈ 1.763, and O(mlogn) mixing for generating random matchings of any graph with bounded degree and m edges.

Optimal mixing of Glauber dynamics: entropy factorization via high-dimensional expansion