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 (Formula presented.) 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 (Formula presented.) 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 (Formula presented.) 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)
Pages (from-to)517-556
Number of pages40
JournalRandom Structures and Algorithms
Volume56
Issue number2
DOIs
StatePublished - Mar 1 2020

Keywords

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

ASJC Scopus subject areas

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

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