Quasi‐polynomial mixing of critical two‐dimensional random cluster models. Issue 2 (13th June 2019) - Gheissari, Reza; Lubetzky, Eyal
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.