Random-Cluster Dynamics on Random Regular Graphs in Tree Uniqueness

Abstract

We establish rapid mixing of the random-cluster Glauber dynamics on random varDelta -regular graphs for all qge 1 and p<p_u(q,varDelta ), where the threshold p_u(q,varDelta ) corresponds to a uniqueness/non-uniqueness phase transition for the random-cluster model on the (infinite) varDelta -regular tree. It is expected that this threshold is sharp, and for q>2 the Glauber dynamics on random varDelta -regular graphs undergoes an exponential slowdown at p_u(q,varDelta ). More precisely, we show that for every qge 1, varDelta ge 3, and p<p_u(q,varDelta ), with probability 1-o(1) over the choice of a random varDelta -regular graph on n vertices, the Glauber dynamics for the random-cluster model has varTheta (n log n) mixing time. As a corollary, we deduce fast mixing of the Swendsen–Wang dynamics for the Potts model on random varDelta -regular graphs for every qge 2, in the tree uniqueness region. Our proof relies on a sharp bound on the “shattering time”, i.e., the number of steps required to break up any configuration into O(log n) sized clusters. This is established by analyzing a delicate and novel iterative scheme to simultaneously reveal the underlying random graph with clusters of the Glauber dynamics configuration on it, at a given time.