Quasi‐polynomial mixing of critical two‐dimensional random cluster models. Issue 2 (13th June 2019) - Gheissari, Reza
Abstract : We study the Glauber dynamics for the random cluster (FK) model on the torus ( Z / n Z ) 2 with parameters ( p, q ), for q ∈ (1, 4] and p the critical point p c . The dynamics is believed to undergo a critical slowdown, with its continuous‐time mixing time transitioning from O ( log n ) for p ≠ p c to a power‐law in n at p = p c . This was verified at p ≠ p c by Blanca and Sinclair, whereas at the critical p = p c, 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 n O ( log n ) at p = p c for all q ∈ (1, 4], where a key ingredient is bounding the number of nested long‐range crossings at criticality.