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Quasi‐polynomial mixing of critical two‐dimensional random cluster models
Author(s) -
Gheissari Reza,
Lubetzky Eyal
Publication year - 2020
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20868
Subject(s) - glauber , mathematics , combinatorics , ising model , mixing (physics) , critical point (mathematics) , upper and lower bounds , torus , polynomial , criticality , power law , slowdown , statistical physics , discrete mathematics , physics , mathematical analysis , geometry , statistics , quantum mechanics , political science , nuclear physics , scattering , law
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.