Corpus ID: 26261692

# 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}
}
• Published 2016
• Mathematics, Physics
• arXiv: Probability
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 $q… Expand #### Figures from this paper Information percolation and cutoff for the random-cluster model • Physics, Mathematics • Random Struct. Algorithms • 2020 This proof proves that for all small enough$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$). We study its Glauber dynamics onExpand The effect of boundary conditions on mixing of 2D Potts models at discontinuous phase transitions • Mathematics, Physics • 2017 We study Swendsen--Wang dynamics for the critical$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}}$is expectedExpand Tunneling behavior of Ising and Potts models on grid graphs • Mathematics • 2017 We consider the ferromagnetic$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 dynamicsExpand 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 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 ΛnExpand 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-temperatureExpand 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 mayExpand 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 parametrizedExpand #### References SHOWING 1-10 OF 28 REFERENCES Quasi-polynomial mixing of the 2D stochastic Ising model with • Mathematics, Physics • 2010 We considerably improve upon the recent result of Martinelli and Toninelli on the mixing time of Glauber dynamics for the 2D Ising model in a box of side$L$at low temperature and with randomExpand On the Mixing Time of the 2D Stochastic Ising Model with “Plus” Boundary Conditions at Low Temperature • Mathematics • 2010 We consider the Glauber dynamics for the 2D Ising model in a box of side L, at inverse temperature β and random boundary conditions τ whose distribution P either stochastically dominates the extremalExpand The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1 • Mathematics, Physics • 2010 We prove a long-standing conjecture on random-cluster models, namely that the critical point for such models with parameter q ≥ 1 on the square lattice is equal to the self-dual point$${p_{sd}(q) =Expand Discontinuity of the phase transition for the planar random-cluster and Potts models with$q>4\$
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We prove that the q-state Potts model and the random-cluster model with cluster weight q > 4 undergo a discontinuous phase transition on the square lattice. More precisely, we show 1. Existence ofExpand
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The random-cluster model has been widely studied as a unifying framework for random graphs, spin systems and electrical networks, but its dynamics have so far largely resisted analysis. In this paperExpand
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This article studies the planar Potts model and its random-cluster representation. We show that the phase transition of the nearest-neighbor ferromagnetic q-state Potts model on Z 2 is continuous forExpand
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The Ising model is widely regarded as the most studied model of spin-systems in statistical physics. The focus of this paper is its dynamic (stochastic) version, the Glauber dynamics, introduced inExpand
Mixing Times of Critical Two‐Dimensional Potts Models
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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-temperatureExpand
Tight bounds for mixing of the Swendsen–Wang algorithm at the Potts transition point
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This work provides the first upper bound of this form for the Swendsen–Wang algorithm, and gives lower bounds for both algorithms which significantly improve the previous lower bounds that were exponential in L/(log L)2. Expand
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We prove Russo-Seymour-Welsh-type uniform bounds on crossing probabilities for the FK Ising (FK percolation with cluster weight q = 2) model at criticality, independent of the boundary conditions.Expand