Rapid Mixing on Random Regular Graphs beyond Uniqueness

2025-04-04

AI Summary

This work investigates the mixing behavior of Glauber dynamics for the hardcore model on random Δ-regular graphs, aiming to uncover fundamental differences between average-case and worst-case sampling complexity. Addressing the long-standing open question—whether rapid mixing persists when λ = O(1/√Δ), far beyond the tree uniqueness threshold λ_c ≈ e/Δ—we establish the first general rapid mixing criterion applicable to downward-closed families. Our method integrates the trickle-down theorem, spectral and entropy stability analysis, and a novel comparison technique linking field dynamics to Glauber dynamics. We rigorously prove that, on random Δ-regular graphs, Glauber dynamics mixes in O(n log n) time—breaking the uniqueness threshold barrier. This result extends to uniform matchings, b-matchings, the random cluster model with q ∈ [0,1), and determinantal point processes, significantly improving sampling efficiency across these models.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

The hardcore model is a fundamental probabilistic model extensively studied in statistical physics, probability theory, and computer science. For graphs of maximum degree $Delta$, 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 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 $Delta$-regular graphs, the Glauber dynamics mixes rapidly with high probability when $lambda = O(1/sqrt{Delta})$, which is significantly beyond the uniqueness threshold $lambda_c(Delta) approx e/Delta$. 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.

Problem

• Studies mixing behavior of Glauber dynamics on random regular graphs.
• Confirms rapid mixing beyond uniqueness threshold for hardcore model.
• Establishes new general criterion for rapid mixing in distributions.

Innovation

• Glauber dynamics mixes rapidly beyond uniqueness threshold
• New criterion for rapid mixing on downward closed sets
• Combines trickle down, spectral stability, and comparison tools