Sampling from the random cluster model on random regular graphs at all temperatures via Glauber dynamics

TL;DR

The paper proves that Glauber dynamics for the RC model on random Δ-regular graphs mixes rapidly when initialized from extreme configurations, for all temperatures and large $q$ (relative to Δ). It combines weak spatial mixing within each phase with local mixing on tree-like neighborhoods and a polymer-expansion framework to bridge local structure to global convergence. For sufficiently large real $q$, the authors obtain $O(n\log n)$ mixing in the disordered phase (β<β_c) from the all-out start and polynomial mixing from the all-in start in the ordered phase (β>β_c), with the critical temperature also addressed. The results imply a polynomial-time, sampling-efficient algorithm for the RC/Potts model across all temperatures (including criticality) under the stated parameter regime, highlighting how strategic initialisation can overcome bottlenecks in multimodal landscapes. The methods extend prior Ising-based approaches to the RC setting through an intricate polymer-based analysis on expander-like graphs and careful local-to-global coupling arguments.

Abstract

We consider the performance of Glauber dynamics for the random cluster model with real parameter $q>1$ and temperature $β>0$. Recent work by Helmuth, Jenssen and Perkins detailed the ordered/disordered transition of the model on random $Δ$-regular graphs for all sufficiently large $q$ and obtained an efficient sampling algorithm for all temperatures $β$ using cluster expansion methods. Despite this major progress, the performance of natural Markov chains, including Glauber dynamics, is not yet well understood on the random regular graph, partly because of the non-local nature of the model (especially at low temperatures) and partly because of severe bottleneck phenomena that emerge in a window around the ordered/disordered transition. Nevertheless, it is widely conjectured that the bottleneck phenomena that impede mixing from worst-case starting configurations can be avoided by initialising the chain more judiciously. Our main result establishes this conjecture for all sufficiently large $q$ (with respect to $Δ$). Specifically, we consider the mixing time of Glauber dynamics initialised from the two extreme configurations, the all-in and all-out, and obtain a pair of fast mixing bounds which cover all temperatures $β$, including in particular the bottleneck window. Our result is inspired by the recent approach of Gheissari and Sinclair for the Ising model who obtained a similar-flavoured mixing-time bound on the random regular graph for sufficiently low temperatures. To cover all temperatures in the RC model, we refine appropriately the structural results of Helmuth, Jenssen and Perkins about the ordered/disordered transition and show spatial mixing properties "within the phase", which are then related to the evolution of the chain.

Theorems & Definitions (72)

• Theorem 1
• Definition 2
• Lemma 3: RCM-Helmuth2020
• proof
• Definition 4
• Theorem 5
• Theorem 6
• Lemma 7: see, e.g., SinclairsGheissari2022,blanca2021random
• Lemma 8: blanca2021random
• Lemma 9