Abstract
We study two widely used algorithms for the Potts model on rectangular subsets of the hypercubic lattice \({\mathbb{Z}^{d}}\)—heat bath dynamics and the Swendsen–Wang algorithm—and prove that, under certain circumstances, the mixing in these algorithms is torpid or slow. In particular, we show that for heat bath dynamics throughout the region of phase coexistence, and for the Swendsen–Wang algorithm at the transition point, the mixing time in a box of side length L with periodic boundary conditions has upper and lower bounds which are exponential in L d-1. 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.
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Aizenman M., Chayes J.T., Chayes L., Newman C.M.: Discontinuity of the magnetization in the one-dimensional 1/|x−y|2 Ising and Potts models. J. Stat. Phys. 50, 1–40 (1988)
Aldous, D., Fill, J.: Reversible Markov chains and random walks on graphs. (in preparation.) Some chapters available at http://stat-www.berkeley.edu/pub/users/aldous/book.html
Alexandrov P.S.: Combinatorial Topology. Dover, New York (1998)
Benjamini I., Mossel E.: On the mixing time of a simple random walk on the super critical percolation cluster. Probab. Theory Relat. Fields 125, 408–420 (2003)
Berger N., Kenyon C., Mossel E., Peres Y.: Glauber dynamics on trees and hyperbolic graphs. Probab. Theory Relat. Fields 131, 311–340 (2005)
Bollobás B., Leader I.: Edge-isoperimetric inequalities in the grid. Combinatorica 11, 299–314 (1991)
Borgs, C.: Statistical Physics Expansion Methods in Combinatorics and Computer Science. CBMS Lectures, Memphis (2003)
Borgs, C., Chayes, J.T., Frieze, A., Kim, J.H., Tetali, P., Vigoda, E., Vu, V.: Torpid mixing of some Monte Carlo Markov chain algorithms in statistical physics. In: Proceedings of the 40th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 218–229. IEEE Computer Society Press (1999)
Borgs V., Imbrie J.: A unified approach to phase diagrams in field theory and statistical mechanics. Commun. Math. Phys. 123, 305–328 (1989)
Borgs C., Kotecký R.: A rigorous theory of finite-size scaling at first-order transitions. J. Stat. Phys. 61, 79–110 (1990)
Borgs C., Kotecký R., Miracle-Solé S.: Finite-size scaling for Potts models. J. Stat. Phys. 62, 529–551 (1991)
Bott, R., Tu, L.W.: Differential forms in algebraic topology. In: Graduate Texts in Mathematics, vol. 82. Springer-Verlag, New York (1986)
Brydges, D.C.: A short course on cluster expansions. Phénomènes critiques, systèmes aléatoires, théories de jauge, Part I, II (Les Houches, 1984), 129–183. North-Holland, Amsterdam (1986)
Cesi F., Maes C., Martinelli F.: Relaxation to equilibrium for two-dimensional disordered Ising systems in the Griffiths phase. Commun. Math. Phys. 189, 323–335 (1997)
Chen, F., Lovász, L., Pak, I.: Unpublished appendix for Lifting Markov chains to speed up mixing. In: Proc. 31st ACM Symp. on Theory of Comp., pp. 275–281 (1999)
Cooper C., Dyer M., Frieze A.M., Rue R.: Mixing properties of the Swendsen-Wang process on the complete graph and narrow grids. J. Math. Phys. 41(3), 1499–1527 (2000)
Cooper C., Frieze A.M.: Mixing properties of the Swendsen-Wang process on classes of graphs. Random Struct. Algorithms 15(3-4), 242–261 (1999)
Dyer M.E., Frieze A.M., Jerrum M.R.: On counting independent sets in sparse graphs. SIAM J. Comput. 31, 1527–1541 (2002)
Edwards R.G., Sokal A.D.: Generalizations of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. Phys. Rev. D38, 2009–2012 (1988)
Fortuin C., Kasteleyn P.: On the random cluster model I: introduction and relation to other models. Physica 57, 536–564 (1972)
Fountoulakis N., Reed B.A.: Faster mixing and small bottlenecks. Probab. Theory Relat. Fields 137, 475–486 (2007)
Gore V., Jerrum M.: The Swendsen-Wang process does not always mix rapidly. J. Stat. Phys. 97(1–2), 67–86 (1999)
Kotecký R., Shlosman S.B.: First-order phase transitions in large entropy lattice models. Commun. Math. Phys. 83, 493–515 (1982)
Lawler G.F., Sokal A.D.: Bounds on the L 2 spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality. Trans. Am. Math. Soc. 309, 557–580 (1988)
Laanait L., Messager A., Miracle-Solé S., Ruiz J., Shlossman S.: Interfaces in the Potts models I: Pirogov-Sinai theory of the Fortuin-Kasteleyn representation. Commun. Math. Phys. 140(1), 81–91 (1991)
Martinelli, F.: Lectures on Glauber dynamics for discrete spin models. In: Lectures on Probability Theory and Statistics (Saint-Flour, 1997). Lecture Notes in Mathematics, vol. 1717, pp. 93–191. Springer, Berlin (1999)
Montenegro R., Tetali P.: Mathematical aspects of mixing times in Markov chains. Found. Trends Theor. Comput. Sci. 1(3), 237–354 (2006)
Pirogov, S.A., Sinai, Ya.G.: Phase diagrams of classical lattice systems. Theor. Math. Phys. 25, 1185–1192 (1975). Phase diagrams of classical lattice systems. (Continuation.) Theor. Math. Phys. 26, 39–49 (1976)
Sinclair A.J., Jerrum M.R.: Approximate counting, uniform generation and rapidly mixing Markov chains. Inf. Comput. 82, 93–133 (1989)
Swendsen R., Wang J.-S.: Non-universal critical dynamics in Monte-Carlo simulation. Phys. Rev. Lett. 58, 86–88 (1987)
Thomas L.: Bound on the mass gap for finite volume stochastic Ising models at low temperature. Commun. Math. Phys. 126, 1–11 (1989)
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P. Tetali research was supported in part by the NSF Grants DMS-9800351, DMS-0401239, DMS-0701043.
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Borgs, C., Chayes, J.T. & Tetali, P. Tight bounds for mixing of the Swendsen–Wang algorithm at the Potts transition point. Probab. Theory Relat. Fields 152, 509–557 (2012). https://doi.org/10.1007/s00440-010-0329-0
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DOI: https://doi.org/10.1007/s00440-010-0329-0