Random-cluster dynamics on random graphs in tree uniqueness
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We establish rapid mixing of the random-cluster Glauber dynamics on random Δ-regular graphs for all q≥ 1 and p<p_u(q,Δ), where the threshold p_u(q,Δ) corresponds to a uniqueness/non-uniqueness phase transition for the random-cluster model on the (infinite) Δ-regular tree. It is expected that for q>2 this threshold is sharp, and the Glauber dynamics on random Δ-regular graphs undergoes an exponential slowdown at p_u(q,Δ). More precisely, we show that for every q≥ 1, Δ≥ 3, and p<p_u(q,Δ), with probability 1-o(1) over the choice of a random Δ-regular graph on n vertices, the Glauber dynamics for the random-cluster model has Θ(n log n) mixing time. As a corollary, we deduce fast mixing of the Swendsen–Wang dynamics for the Potts model on random Δ-regular graphs for every q≥ 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.
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