### 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 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.

Original language | English (US) |
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Journal | Random Structures and Algorithms |

DOIs | |

State | Published - Jan 1 2019 |

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### 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.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Quasi-polynomial mixing of critical two-dimensional random cluster models

AU - Gheissari, Reza

AU - Lubetzky, Eyal

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - critical phenomena

KW - Glauber dynamics

KW - mixing time

KW - random cluster model

UR - http://www.scopus.com/inward/record.url?scp=85067390259&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85067390259&partnerID=8YFLogxK

U2 - 10.1002/rsa.20868

DO - 10.1002/rsa.20868

M3 - Article

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

ER -