We establish rapid mixing of the randomcluster Glauber dynamics on random
We study the performance of Markov chains for the
 NSFPAR ID:
 10396273
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Communications in Mathematical Physics
 Volume:
 401
 Issue:
 1
 ISSN:
 00103616
 Format(s):
 Medium: X Size: p. 185225
 Size(s):
 ["p. 185225"]
 Sponsoring Org:
 National Science Foundation
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Abstract regular graphs for all$$\varDelta $$ $\Delta $ and$$q\ge 1$$ $q\ge 1$ , where the threshold$$p $p<{p}_{u}(q,\Delta )$ corresponds to a uniqueness/nonuniqueness phase transition for the randomcluster model on the (infinite)$$p_u(q,\varDelta )$$ ${p}_{u}(q,\Delta )$ regular tree. It is expected that this threshold is sharp, and for$$\varDelta $$ $\Delta $ the Glauber dynamics on random$$q>2$$ $q>2$ regular graphs undergoes an exponential slowdown at$$\varDelta $$ $\Delta $ . More precisely, we show that for every$$p_u(q,\varDelta )$$ ${p}_{u}(q,\Delta )$ ,$$q\ge 1$$ $q\ge 1$ , and$$\varDelta \ge 3$$ $\Delta \ge 3$ , with probability$$p $p<{p}_{u}(q,\Delta )$ over the choice of a random$$1o(1)$$ $1o\left(1\right)$ regular graph on$$\varDelta $$ $\Delta $n vertices, the Glauber dynamics for the randomcluster model has mixing time. As a corollary, we deduce fast mixing of the Swendsen–Wang dynamics for the Potts model on random$$\varTheta (n \log n)$$ $\Theta (nlogn)$ regular graphs for every$$\varDelta $$ $\Delta $ , 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$$q\ge 2$$ $q\ge 2$ 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.$$O(\log n)$$ $O(logn)$ 
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