Optimal Mixing of Glauber Dynamics: Entropy Factorization via High-Dimensional Expansion
We consider the Glauber dynamics (also called Gibbs sampling) for sampling from a discrete distribution in high-dimensional space (e.g., selecting a uniformly random legal coloring or independent set in a given graph, or selecting a "typical" state in a given statistical physics model, like spin systems). In these probabilistic algorithms, each step picks uniformly at random a single variable, and updates it conditional on all other variables. Note that, if the number of variables (= the dimension) is n, such MCMC (Monte Carlo Markov Chain) algorithms perform a random walk on an exponential (in n) size space.
We show an optimal mixing time bound for the Glauber dynamics 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 for graphical models/spin systems on any n-vertex graph of bounded degree, when the maximum eigenvalue of an associated influence matrix is bounded. Our proof approach combines classic tools of entropy tensorization/factorization and recent developments of high-dimensional expanders.
Based on joint work with Kuikui Liu and Eric Vigoda.