Rapid Mixing on Random Regular Graphs beyond Uniqueness
Xiaoyu Chen, Zejia Chen, Zongchen Chen, Yitong Yin, Xinyuan Zhang
TL;DR
This work analyzes the Glauber dynamics for the hardcore model and related distributions on combinatorial structures, focusing on beyond-worst-case (average-case) instances such as random $\Delta$-regular graphs. It introduces a general criterion for rapid mixing on downward-closed sets via field dynamics, backed by a trickle-down theorem and a novel comparison to Glauber dynamics, and it shows that rapid mixing holds for $\lambda = O(1/\sqrt{\Delta})$ in random regular graphs, far beyond the tree-uniqueness threshold. The results extend to a broad family of models, including matching and $b$-matching, the random-cluster model on matroids, Holant problems with log-concave signatures, and determinantal point processes, yielding improved mixing bounds and establishing log-concavity of generating polynomials. The findings reveal a sharp separation between worst-case hardness and average-case tractability, and they provide a versatile, general toolkit for proving rapid mixing across numerous anti-ferromagnetic and combinatorial models with practical implications for sampling and counting.
Abstract
The hardcore model is a fundamental probabilistic model extensively studied in statistical physics, probability theory, and computer science. For graphs of maximum degree $Δ$, a well-known computational phase transition occurs at the tree-uniqueness threshold $λ_c(Δ) = \frac{(Δ-1)^{Δ-1}}{(Δ-2)^Δ}$, where the mixing behavior of the Glauber dynamics (a simple Markov chain) undergoes a sharp transition. 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 $Δ$-regular graphs, the Glauber dynamics mixes rapidly with high probability when $λ= O(1/\sqrtΔ)$, which is significantly beyond the uniqueness threshold $λ_c(Δ) \approx e/Δ$. 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 $q \in [0,1)$, 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.