Mixing Times of Critical Two‐Dimensional Potts Models

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 bottlenecks) are all expected to undergo “critical slowdowns” in the presence of periodic boundary conditions: the inverse spectral gap, which in the subcritical regime is O(1), should at criticality be polynomial in n for 1   4 in accordance with the predicted discontinuous phase transition. This was confirmed for q = 2 (the Ising model) by the second author and Sly, and for sufficiently large q by Borgs et al. Here we show that the following holds for the critical Potts model on the torus: for q=3, the inverse gap of Glauber dynamics is nO(1); for q = 4, it is at most nO(log n); and for every q > 4 in the phase-coexistence regime, the inverse gaps of both Glauber dynamics and Swendsen-Wang dynamics are exponential in n. For free or monochromatic boundary conditions and large q, we show that the dynamics at criticality is faster than on the torus (unlike the Ising model where free/periodic boundary conditions induce similar dynamical behavior at all temperatures): the inverse gap of Swendsen-Wang dynamics is exp(no(1)). © 2017 Wiley Periodicals, Inc.

[1]  Alistair Sinclair,et al.  Dynamics for the Mean-field Random-cluster Model , 2014, APPROX-RANDOM.

[2]  C. H. C. Little,et al.  Evolutionary Families of Sets , 2000 .

[3]  Wang,et al.  Nonuniversal critical dynamics in Monte Carlo simulations. , 1987, Physical review letters.

[4]  Lahoussine Laanait,et al.  Interfaces in the Potts model I: Pirogov-Sinai theory of the Fortuin-Kasteleyn representation , 1991 .

[5]  K. Alexander,et al.  On weak mixing in lattice models , 1998 .

[6]  Alistair Sinclair,et al.  Improved Bounds for Mixing Rates of Markov Chains and Multicommodity Flow , 1992, Combinatorics, Probability and Computing.

[7]  André Pönitz,et al.  Improved Upper Bounds for Self-Avoiding Walks in ${\bf Z}^{d}$ , 2000 .

[8]  P. Hohenberg,et al.  Theory of Dynamic Critical Phenomena , 1977 .

[9]  C. Fortuin,et al.  Correlation inequalities on some partially ordered sets , 1971 .

[10]  Mario Ullrich,et al.  Comparison of Swendsen‐Wang and heat‐bath dynamics , 2011, Random Struct. Algorithms.

[11]  A. Sokal,et al.  Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. , 1988, Physical review. D, Particles and fields.

[12]  Mario Ullrich.  Rapid mixing of Swendsen–Wang dynamics in two dimensions , 2012, 1212.4908.

[13]  H. Duminil-Copin,et al.  The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1 , 2010, Probability Theory and Related Fields.

[14]  Kent Bækgaard Lauritsen,et al.  Critical exponents from power spectra , 1993 .

[15]  Allan Sly,et al.  Critical Ising on the Square Lattice Mixes in Polynomial Time , 2010, 1001.1613.

[16]  Mark Jerrum,et al.  Approximating the Permanent , 1989, SIAM J. Comput..

[17]  F. Toninelli,et al.  On the Mixing Time of the 2D Stochastic Ising Model with “Plus” Boundary Conditions at Low Temperature , 2009, 0905.3040.

[18]  Allan Sly,et al.  Quasi-polynomial mixing of the 2D stochastic Ising model with , 2010, 1012.1271.

[19]  F. Martinelli,et al.  Approach to equilibrium of Glauber dynamics in the one phase region , 1994 .

[20]  J. Chayes,et al.  Exponential decay of connectivities in the two-dimensional ising model , 1987 .

[21]  F. Martinelli,et al.  Approach to equilibrium of Glauber dynamics in the one phase region , 1994 .

[22]  Senya Shlosman,et al.  Interfaces in the Potts model II: Antonov's rule and rigidity of the order disorder interface , 1991 .

[23]  Alan M. Frieze,et al.  Torpid mixing of some Monte Carlo Markov chain algorithms in statistical physics , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[24]  Li,et al.  Rigorous lower bound on the dynamic critical exponent of some multilevel Swendsen-Wang algorithms. , 1991, Physical review letters.

[25]  H. Duminil-Copin,et al.  Continuity of the Phase Transition for Planar Random-Cluster and Potts Models with 1≤q≤4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} , 2016, Communications in Mathematical Physics.

[26]  Reza Gheissari,et al.  Quasi-polynomial mixing of critical 2D random cluster models , 2016 .

[27]  Jean Ruiz,et al.  Phases coexistence and surface tensions for the potts model , 1986 .

[28]  Alistair Sinclair,et al.  Random-cluster dynamics in Z 2 , 2016, SODA 2016.

[29]  Pierre Nolin,et al.  Connection probabilities and RSW‐type bounds for the two‐dimensional FK Ising model , 2011 .

[30]  R. Glauber.  Time‐Dependent Statistics of the Ising Model , 1963 .

[31]  Senya Shlosman,et al.  First-order phase transitions in large entropy lattice models , 1982 .

[32]  Y. Peres,et al.  Can Extra Updates Delay Mixing? , 2011, 1112.0603.

[33]  P. Diaconis,et al.  COMPARISON THEOREMS FOR REVERSIBLE MARKOV CHAINS , 1993 .

[34]  F. Martinelli,et al.  For 2-D lattice spin systems weak mixing implies strong mixing , 1994 .

[35]  P. Diaconis,et al.  Geometric Bounds for Eigenvalues of Markov Chains , 1991 .

[36]  M. Jerrum,et al.  The Swendsen–Wang Process Does Not Always Mix Rapidly , 1999 .

[37]  Giovanni Ossola,et al.  Dynamic Critical Behavior of the Chayes–Machta Algorithm for the Random-Cluster Model, I. Two Dimensions , 2011, 1105.0373.

[38]  Christian Borgs,et al.  Tight bounds for mixing of the Swendsen–Wang algorithm at the Potts transition point , 2010, ArXiv.

[39]  R. Baxter,et al.  Magnetisation discontinuity of the two-dimensional Potts model , 1982 .

[40]  F. Martinelli.  On the two-dimensional dynamical Ising model in the phase coexistence region , 1994 .

[41]  F. Y. Wu.  The Potts model , 1982 .

[42]  Jonathan Machta,et al.  Graphical representations and cluster algorithms I. Discrete spin systems , 1997 .

[43]  F. Martinelli.  Lectures on Glauber dynamics for discrete spin models , 1999 .

[44]  S. Smirnov.  Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model , 2007, 0708.0039.

[45]  Paul Cuff,et al.  Glauber Dynamics for the Mean-Field Potts Model , 2012, 1204.4503.

[46]  F. Martinelli,et al.  On the two-dimensional stochastic Ising model in the phase coexistence region near the critical point , 1996 .

[47]  Eric Vigoda,et al.  Swendsen‐Wang algorithm on the mean‐field Potts model , 2015, APPROX-RANDOM.

[48]  Vladas Sidoravicius,et al.  Continuity of the Phase Transition for Planar Random-Cluster and Potts Models with $${1 \le q \le 4}$$1≤q≤4 , 2015, 1505.04159.