The Annals of Applied Probability

Mixing time for the solid-on-solid model

Fabio Martinelli and Alistair Sinclair

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We analyze the mixing time of a natural local Markov chain (the Glauber dynamics) on configurations of the solid-on-solid model of statistical physics. This model has been proposed, among other things, as an idealization of the behavior of contours in the Ising model at low temperatures. Our main result is an upper bound on the mixing time of Õ(n3.5), which is tight within a factor of Õ(√n). The proof, which in addition gives some insight into the actual evolution of the contours, requires the introduction of a number of novel analytical techniques that we conjecture will have other applications.

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Ann. Appl. Probab., Volume 22, Number 3 (2012), 1136-1166.

First available in Project Euclid: 18 May 2012

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs

Solid-on-solid model Markov chain Monte Carlo censoring Glauber dynamics mixing time monotonicity Ising model


Martinelli, Fabio; Sinclair, Alistair. Mixing time for the solid-on-solid model. Ann. Appl. Probab. 22 (2012), no. 3, 1136--1166. doi:10.1214/11-AAP791.

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  • [1] Aldous, D. (1983). Random walks on finite groups and rapidly mixing Markov chains. In Seminar on Probability, XVII. Lecture Notes in Math. 986 243–297. Springer, Berlin.
  • [2] Berger, N., Kenyon, C., Mossel, E. and Peres, Y. (2005). Glauber dynamics on trees and hyperbolic graphs. Probab. Theory Related Fields 131 311–340.
  • [3] Bianchi, A. (2008). Glauber dynamics on nonamenable graphs: Boundary conditions and mixing time. Electron. J. Probab. 13 1980–2013.
  • [4] Bodineau, T. and Martinelli, F. (2002). Some new results on the kinetic Ising model in a pure phase. J. Stat. Phys. 109 207–235.
  • [5] Caputo, P., Martinelli, F. and Toninelli, F. L. (2008). On the approach to equilibrium for a polymer with adsorption and repulsion. Electron. J. Probab. 13 213–258.
  • [6] Caputo, P., Martinelli, F. and Toninelli, F. L. (2011). Mixing times of monotone surfaces and SOS interfaces: A mean curvature approach. Preprint. Available at
  • [7] Cesi, F. (2001). Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields. Probab. Theory Related Fields 120 569–584.
  • [8] Chayes, L., Schonmann, R. H. and Swindle, G. (1995). Lifshitz’ law for the volume of a two-dimensional droplet at zero temperature. J. Stat. Phys. 79 821–831.
  • [9] Diaconis, P. and Saloff-Coste, L. (1993). Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3 696–730.
  • [10] Dobrushin, R., Kotecký, R. and Shlosman, S. (1992). Wulff Construction: A Global Shape from Local Interaction. Translations of Mathematical Monographs 104. Amer. Math. Soc., Providence, RI.
  • [11] Dunlop, F. M., Ferrari, P. A. and Fontes, L. R. G. (2002). A dynamic one-dimensional interface interacting with a wall. J. Stat. Phys. 107 705–727.
  • [12] Funaki, T. (2005). Stochastic interface models. In Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1869 103–274. Springer, Berlin.
  • [13] Giacomin, G. (2007). Random Polymer Models. Imperial College Press, London.
  • [14] Huse, D. and Fisher, D. (1987). Dynamics of droplet fluctuations in pure and random Ising systems. Phys. Rev. B 35 6841–6846.
  • [15] Levin, D. A., Luczak, M. J. and Peres, Y. (2010). Glauber dynamics for the mean-field Ising model: Cut-off, critical power law, and metastability. Probab. Theory Related Fields 146 223–265.
  • [16] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI.
  • [17] Luby, M., Randall, D. and Sinclair, A. (2001). Markov chain algorithms for planar lattice structures. SIAM J. Comput. 31 167–192 (electronic).
  • [18] Martinelli, F. (1999). Lectures on Glauber dynamics for discrete spin models. In Lectures on Probability Theory and Statistics (Saint-Flour, 1997). Lecture Notes in Math. 1717 93–191. Springer, Berlin.
  • [19] Martinelli, F. and Olivieri, E. (1994). Approach to equilibrium of Glauber dynamics in the one phase region. I. The attractive case. Comm. Math. Phys. 161 447–486.
  • [20] Martinelli, F., Sinclair, A. and Weitz, D. (2004). Glauber dynamics on trees: Boundary conditions and mixing time. Comm. Math. Phys. 250 301–334.
  • [21] Martinelli, F. and Toninelli, F. L. (2010). On the mixing time of the 2D stochastic Ising model with “plus” boundary conditions at low temperature. Comm. Math. Phys. 296 175–213.
  • [22] Peres, Y. (2005). Mixing for Markov chains and spin systems. Lecture Notes for PIMS Summer School at UBC, August 2005. Available at
  • [23] Posta, G. (1995/97). Spectral gap for an unrestricted Kawasaki type dynamics. ESAIM Probab. Stat. 1 145–181 (electronic).
  • [24] Privman, V. and Švrakić, N. M. (1988). Difference equations in statistical mechanics. II. Solid-on-solid models in two dimensions. J. Stat. Phys. 51 1111–1126.
  • [25] Privman, V. and Švrakić, N. M. (1989). Line interfaces in two dimensions: Solid-on-solid models. In Directed Models of Polymers, Interfaces, and Clusters: Scaling and Finite-Size Properties. Lecture Notes in Physics 338 32–60. Springer, Berlin.
  • [26] Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9 223–252.
  • [27] Randall, D. and Tetali, P. (2000). Analyzing Glauber dynamics by comparison of Markov chains. J. Math. Phys. 41 1598–1615.
  • [28] Spitzer, F. (1964). Principles of Random Walk. Van Nostrand, Princeton, NJ.
  • [29] Stroock, D. W. and Zegarliński, B. (1992). The logarithmic Sobolev inequality for discrete spin systems on a lattice. Comm. Math. Phys. 149 175–193.
  • [30] Wilson, D. B. (2004). Mixing times of Lozenge tiling and card shuffling Markov chains. Ann. Appl. Probab. 14 274–325.