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

Reza Gheissari; Eyal Lubetzky

doi:10.1002/rsa.20868

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

Reza Gheissari, Eyal Lubetzky

Mathematics

Research output: Contribution to journal › Article › peer-review

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 p_c. The dynamics is believed to undergo a critical slowdown, with its continuous-time mixing time transitioning from (Formula presented.) 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 (Formula presented.) at p = p_c for all q ∈ (1,4], where a key ingredient is bounding the number of nested long-range crossings at criticality.

Original language	English (US)
Pages (from-to)	517-556
Number of pages	40
Journal	Random Structures and Algorithms
Volume	56
Issue number	2
DOIs	https://doi.org/10.1002/rsa.20868
State	Published - Mar 1 2020

Keywords

Glauber dynamics
critical phenomena
mixing time
random cluster model

ASJC Scopus subject areas

Software
General Mathematics
Computer Graphics and Computer-Aided Design
Applied Mathematics

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10.1002/rsa.20868

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title = "Quasi-polynomial mixing of critical two-dimensional random cluster models",

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.",

keywords = "Glauber dynamics, critical phenomena, mixing time, random cluster model",

author = "Reza Gheissari and Eyal Lubetzky",

note = "Funding Information: information This research was supported by the NSF; DMS-1507019 and DMS-1513403.The authors thank the anonymous referees for their helpful suggestions. R.G. was supported in part by NSF grant DMS-1507019. E.L. was supported in part by NSF grant DMS-1513403. Publisher Copyright: {\textcopyright} 2019 Wiley Periodicals, Inc.",

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N1 - Funding Information: information This research was supported by the NSF; DMS-1507019 and DMS-1513403.The authors thank the anonymous referees for their helpful suggestions. R.G. was supported in part by NSF grant DMS-1507019. E.L. was supported in part by NSF grant DMS-1513403. Publisher Copyright: © 2019 Wiley Periodicals, Inc.

PY - 2020/3/1

Y1 - 2020/3/1

N2 - 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.

AB - 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.

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

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Keywords

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