# Quasi-polynomial mixing of critical 2D random cluster models

@article{Gheissari2016QuasipolynomialMO, title={Quasi-polynomial mixing of critical 2D random cluster models}, author={Reza Gheissari and Eyal Lubetzky}, journal={arXiv: Probability}, year={2016} }

We study the Glauber dynamics for the random cluster (FK) model on the torus $(\mathbb{Z}/n\mathbb{Z})^2$ with parameters $(p,q)$, for $q \in (1,4]$ 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 $p\neq p_c$ to a power-law in $n$ at $p=p_c$. This was verified at $p\neq p_c$ by Blanca and Sinclair, whereas at the critical $p=p_c$, with the exception of the special integer points $q… Expand

#### 11 Citations

Information percolation and cutoff for the random-cluster model

- Physics, Mathematics
- Random Struct. Algorithms
- 2020

This proof proves that for all small enough $p$ (depending on the dimension) and any $q>1$, the FK-dynamics exhibits the cutoff phenomenon at $\lambda_{\infty}^{-1}\log n$ with a window size $O(\log\log n)$. Expand

Random-cluster dynamics in $\mathbb{Z}^{2}$: Rapid mixing with general boundary conditions

- Mathematics, Physics
- 2018

The random-cluster model with parameters $(p,q)$ is a random graph model that generalizes bond percolation ($q=1$) and the Ising and Potts models ($q\geq 2$). We study its Glauber dynamics on… Expand

The effect of boundary conditions on mixing of 2D Potts models at discontinuous phase transitions

- Mathematics, Physics
- 2017

We study Swendsen--Wang dynamics for the critical $q$-state Potts model on the square lattice. For $q=2,3,4$, where the phase transition is continuous, the mixing time $t_{\textrm{mix}}$ is expected… Expand

Tunneling behavior of Ising and Potts models on grid graphs

- Mathematics
- 2017

We consider the ferromagnetic $q$-state Potts model with zero external field in a finite volume and assume that the stochastic evolution of this system is described by a Glauber-type dynamics… Expand

Random-Cluster Dynamics in Z2: Rapid Mixing with General Boundary Conditions

- Mathematics, Computer Science
- APPROX-RANDOM
- 2019

It is proved that when q > 1 and p 6= pc(q), the Glauber dynamics on Λn mixes in optimal O(n logn) time, which is polynomial in n for every boundary condition that is realizable as a configuration on Z \Λn. Expand

PR ] 6 M ay 2 01 9 Random-cluster dynamics in Z 2 : rapid mixing with general boundary conditions

- 2019

The random-cluster model with parameters (p, q) is a random graph model that generalizes bond percolation (q = 1) and the Ising and Potts models (q ≥ 2). We study its Glauber dynamics on n×n boxes Λn… Expand

Mixing Times of Critical Two‐Dimensional Potts Models

- Mathematics
- 2018

We study dynamical aspects of the q-state Potts model on an n × n box at its critical βc(q). Heat-bath Glauber dynamics and cluster dynamics such as Swendsen–Wang (that circumvent low-temperature… Expand

Swendsen-Wang algorithm on the mean-field Potts model

- Physics, Computer Science
- Random Struct. Algorithms
- 2019

The Swendsen-Wang algorithm which is a Markov chain that utilizes the random cluster representation for the ferromagnetic Potts model to recolor large sets of vertices in one step and potentially overcomes obstacles that inhibit single-site Glauber dynamics is analyzed. Expand

Renormalization of Crossing Probabilities in the Planar Random-Cluster Model

- Mathematics, Physics
- 2019

The study of crossing probabilities - i.e. probabilities of existence of paths crossing rectangles - has been at the heart of the theory of two-dimensional percolation since its beginning. They may… Expand

Tunneling behavior of Ising and Potts models in the low-temperature regime

- Mathematics, Physics
- Stochastic Processes and their Applications
- 2019

Abstract We consider the ferromagnetic q -state Potts model with zero external field in a finite volume and assume that its stochastic evolution is described by a Glauber-type dynamics parametrized… Expand

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