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

Reza Gheissari, Eyal Lubetzky

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
JournalRandom Structures and Algorithms
DOIs
StatePublished - Jan 1 2019

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Random-cluster Model
Potts model
Polynomials
Upper bound
Glauber Dynamics
Mixing Time
Integer Points
Polynomial
Potts Model
Criticality
Ising Model
Exception
Continuous Time
Torus
Power Law
Range of data
Model

Keywords

  • critical phenomena
  • Glauber dynamics
  • mixing time
  • random cluster model

ASJC Scopus subject areas

  • Software
  • Mathematics(all)
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics

Cite this

Quasi-polynomial mixing of critical two-dimensional random cluster models. / Gheissari, Reza; Lubetzky, Eyal.

In: Random Structures and Algorithms, 01.01.2019.

Research output: Contribution to journalArticle

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