2016/11/03 by Reza Gheissari, Gheissari, Reza, Eyal Lubetzky +1 · 8 citations
Mathematics · Physics and Astronomy · #60K35 #82B20 #82B27 #82C20 #Cluster (spacecraft) #Combinatorics #Critical point (mathematics) #Discrete mathematics #FOS: Mathematics #FOS: Physical sciences #Geometry #Glauber #Ising model #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Mathematical analysis #Mathematics #Mixing (physics) #Physics #Polynomial #Probability (math.PR) #Quantum mechanics #Statistical physics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Torus #Upper and lower bounds #math-ph #math.MP #math.PR #msc:60K35 #msc:82B20 #msc:82B27 #msc:82C20
paper · pdf · doi:10.48550/arxiv.1611.01147
published in arXiv (Cornell University) (Cornell University) · 39 pages, 8 figures
openalex publication_date 2016/11/03 · arxiv created 2019/03/30 · arxiv updated 2019/04/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(log n) 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(log n) at p=pc for all q∈ (1,4], where a key ingredient is bounding the number of nested long-range crossings at criticality.