Rapid Mixing on Random Regular Graphs beyond Uniqueness

The hardcore model is a fundamental probabilistic model extensively studied in statistical physics, probability theory, and computer science. It defines a Gibbs distribution over independent sets of a given graph, parameterized by a vertex activity λ > 0. For graphs of maximum degree ∆, a well-known computational phase transition occurs at the tree-uniqueness threshold ${\lambda _c}(\Delta ) = \frac{{{{(\Delta - 1)}^{\Delta - 1}}}}{{{{(\Delta - 2)}^\Delta }}}$, where the mixing behavior of the Glauber dynamics (a simple Markov chain) undergoes a sharp transition: it mixes in nearly linear time for λc(∆), in polynomial but super-linear time at λ = λc(∆), and experiences exponential slowdown for λ > λc(∆).It is conjectured that random regular graphs exhibit different mixing behavior, with the slowdown occurring far beyond the uniqueness threshold. We confirm this conjecture by showing that, for the hardcore model on random ∆-regular graphs, the Glauber dynamics mixes rapidly with high probability when $\lambda = O\left( {1/\sqrt \Delta } \right)$, which is significantly beyond the uniqueness threshold λc(∆) ≈ e/∆. Our result establishes a sharp distinction between the hardcore model on worst-case and beyond-worst-case instances, showing that the worst-case and average-case complexities of sampling and counting are fundamentally different.This result of rapid mixing on random instances follows from a new criterion we establish for rapid mixing of Glauber dynamics for any distribution supported on a downward closed set family. Our criterion is simple, general, and easy to check. In addition to proving new mixing conditions for the hardcore model, we also establish improved mixing time bounds for sampling uniform matchings or b-matchings on graphs, the random cluster model on matroids with q ∈ [0,1), and the determinantal point process. Our proof of this new criterion for rapid mixing combines and generalizes several recent tools in a novel way, including a trickle-down theorem for field dynamics, spectral/entropic stability, and a new comparison result between field dynamics and Glauber dynamics.