Quasi‐polynomial mixing of critical two‐dimensional random cluster models

We study the Glauber dynamics for the random cluster (FK) model on the torus (Z/nZ)2 with parameters (p,q), for q ∈ (1,4] and p the critical point pc. The dynamics is believed to undergo a critical slowdown, with its continuous‐time mixing time transitioning from O(logn) for p ≠ pc to a power‐law in...

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Bibliographic Details
Published in:Random structures & algorithms 2020-03, Vol.56 (2), p.517-556
Main Authors: Gheissari, Reza, Lubetzky, Eyal
Format: Article
Language:English
Subjects:
Clusters
critical phenomena
Critical point
Glauber dynamics
Ising model
mixing time
Polynomials
random cluster model
Toruses
Upper bounds
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Summary:We study the Glauber dynamics for the random cluster (FK) model on the torus (Z/nZ)2 with parameters (p,q), for q ∈ (1,4] and p the critical point pc. The dynamics is believed to undergo a critical slowdown, with its continuous‐time mixing time transitioning from O(logn) for p ≠ pc to a power‐law in n at p = pc. This was verified at p ≠ pc by Blanca and Sinclair, whereas at the critical p = pc, with the exception of the special integer points q = 2,3,4 (where the model corresponds to the Ising/Potts models) the best‐known upper bound on mixing was exponential in n. Here we prove an upper bound of nO(logn) at p = pc for all q ∈ (1,4], where a key ingredient is bounding the number of nested long‐range crossings at criticality.
ISSN:1042-9832
1098-2418
DOI:10.1002/rsa.20868