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

Abstract

We study the Glauber dynamics for the random cluster (FK) model on the torus 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 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 at p = pc for all q ∈ (1,4], where a key ingredient is bounding the number of nested long‐range crossings at criticality.