Random-Cluster Dynamics on Random Regular Graphs in Tree Uniqueness

Abstract

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

Metastability of the Potts ferromagnet on random regular graphs

“…Discussion. Our slow mixing result for Glauber dynamics when β > β u (Theorem 1.1) significantly improves upon previous results of Bordewich, Greenhill and Patel [11] that applied to β > β u + Θ q (1); in fact, it complements the recent fast mixing result of Blanca and Gheissari [8] on the random d-regular graph that applies to all β < β u , leaving therefore only open the mixing time at the critical case β = β u (which is believed to be polynomial in n).…”

“…Extrapolating from the mean-field case (see discussion below), it is natural to conjecture that this slow mixing result is best-possible, i.e., for β / ∈ (β u , β h ), SW mixes rapidly on the random regular graph. Note, the result of [8] already implies a polynomial bound on the mixing time of SW when β < β u (due to comparison results by Ullrich that apply to general graphs [45]).…”

Finite-size scaling, phase coexistence, and algorithms for the random cluster model on random graphs

“…The next theorem takes a step towards confirming that the mean-field picture is correct for random regular graphs by proving slow mixing in an interval. Previously slow mixing was only known at criticality, consistent with both lattice and mean-field-type behavior [32,4]. Theorem 6.…”

“…It has been conjectured that the Swendsen-Wang dynamics for random regular graphs exhibit mean-field behavior, mixing exponentially exponentially slowly for q > 2 and β in the entire interval (β u , β ⋆ u ). See [4] for a discussion. The next theorem takes a step towards confirming that the mean-field picture is correct for random regular graphs by proving slow mixing in an interval.…”

“…The most relevant results for the present paper are that, for the case of random ∆regular graphs, Blanca, Galanis, Goldberg, Štefankovič, Vigoda, and Yang [3] give an efficient sampling algorithm for the Potts and random cluster models when the temperature is above the uniqueness threshold of the infinite ∆-regular tree, i.e., β < β u (T ∆ ); Blanca and Gheissari then showed O(n log n) mixing of the random cluster dynamics for β < β u [4]. In a more general setting, Bordewich, Greenhill, and Patel showed the Glauber dynamics for the Potts model on graphs of maximum degree at most ∆ mix rapidly for β ≤ (1 + o q (1)) log q/(∆ − 1) and showed slow mixing of the Glauber dynamics on random regular graphs for β ≥ (1 + o q (1)) log q/(∆ − 1 − 1/(∆ − 1)) [8].…”