Sampling from Potts on random graphs of unbounded degree via random-cluster dynamics

Abstract

We consider the problem of sampling from the ferromagnetic Potts and random-cluster models on a general family of random graphs via the Glauber dynamics for the random-cluster model. The randomcluster model is parametrized by an edge probability p P p0, 1q and a cluster weight q ą 0. We establish that for every q ě 1, the random-cluster Glauber dynamics mixes in optimal Θpn log nq steps on n-vertex random graphs having a prescribed degree sequence with bounded average branching γ, throughout the uniqueness regime p ă pupq, γq. Notably, this uniqueness threshold does not decay with the maximum degree and only depends on the average degree. In particular, pupq, γq is expected to be sharp, in that for general q and γ, the random-cluster Glauber dynamics should slow down dramatically at pupq, γq.The family of random graph models we consider include the Erdős-Rényi random graph Gpn, γ{nq, and so we provide the first polynomial-time sampling algorithm for the ferromagnetic Potts model on the Erdős-Rényi random graphs that works for all q in the full uniqueness regime. We accompany our results with mixing time lower bounds (exponential in the maximum degree) for the Potts Glauber dynamics, in the same settings where our Θpn log nq bounds for the random-cluster Glauber dynamics apply. This reveals a significant computational advantage of random-cluster based algorithms for sampling from the Potts Gibbs distribution at high temperatures in the presence of high-degree vertices.