TY - JOUR

T1 - Random-Cluster Dynamics on Random Regular Graphs in Tree Uniqueness

AU - Blanca, Antonio

AU - Gheissari, Reza

N1 - Funding Information:
The authors thank the anonymous referee for detailed comments on the manuscript. The research of A.B. was supported in part by NSF Grant CCF-1850443. R.G. thanks the Miller Institute for Basic Research in Science for its support.
Publisher Copyright:
© 2021, The Author(s).

PY - 2021/9

Y1 - 2021/9

N2 - We establish rapid mixing of the random-cluster Glauber dynamics on random Δ-regular graphs for all q≥ 1 and p< pu(q, Δ) , where the threshold pu(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 this threshold is sharp, and for q> 2 the Glauber dynamics on random Δ-regular graphs undergoes an exponential slowdown at pu(q, Δ). More precisely, we show that for every q≥ 1 , Δ≥ 3 , and p< pu(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(nlog 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.

AB - We establish rapid mixing of the random-cluster Glauber dynamics on random Δ-regular graphs for all q≥ 1 and p< pu(q, Δ) , where the threshold pu(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 this threshold is sharp, and for q> 2 the Glauber dynamics on random Δ-regular graphs undergoes an exponential slowdown at pu(q, Δ). More precisely, we show that for every q≥ 1 , Δ≥ 3 , and p< pu(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(nlog 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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U2 - 10.1007/s00220-021-04093-z

DO - 10.1007/s00220-021-04093-z

M3 - Article

AN - SCOPUS:85104833502

VL - 386

SP - 1243

EP - 1287

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -