Mixing time for the solid-on-solid model
Fabio Martinelli and Alistair Sinclair
Source: Ann. Appl. Probab. Volume 22, Number 3
(2012), 1136-1166.
Abstract
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.
First Page:
Show
Hide
Keywords: Solid-on-solid model; Markov chain Monte Carlo; censoring; Glauber dynamics; mixing time; monotonicity; Ising model
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aoap/1337347541
Digital Object Identifier: doi:10.1214/11-AAP791
Zentralblatt MATH identifier: 06053741
Mathematical Reviews number (MathSciNet): MR2977988
References
[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.
Mathematical Reviews (MathSciNet): MR770418
Zentralblatt MATH: 0514.60067
[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.
Mathematical Reviews (MathSciNet): MR2123248
Zentralblatt MATH: 1075.60003
Digital Object Identifier: doi:10.1007/s00440-004-0369-4
[3] Bianchi, A. (2008). Glauber dynamics on nonamenable graphs: Boundary conditions and mixing time. Electron. J. Probab. 13 1980–2013.
Mathematical Reviews (MathSciNet): MR2453553
Zentralblatt MATH: 1187.82073
Digital Object Identifier: doi:10.1214/EJP.v13-574
[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.
Mathematical Reviews (MathSciNet): MR1927919
Zentralblatt MATH: 1027.82028
Digital Object Identifier: doi:10.1023/A:1019939712267
[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.
Mathematical Reviews (MathSciNet): MR2386733
Zentralblatt MATH: 1191.60110
Digital Object Identifier: doi:10.1214/EJP.v13-486
[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 http://arxiv.org/abs/1101.4190.
[7] Cesi, F. (2001). Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields. Probab. Theory Related Fields 120 569–584.
Mathematical Reviews (MathSciNet): MR1853483
Zentralblatt MATH: 1086.82002
Digital Object Identifier: doi:10.1007/PL00008792
[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.
Mathematical Reviews (MathSciNet): MR1330364
Zentralblatt MATH: 1081.82544
Digital Object Identifier: doi:10.1007/BF02181205
[9] Diaconis, P. and Saloff-Coste, L. (1993). Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3 696–730.
Mathematical Reviews (MathSciNet): MR1233621
Zentralblatt MATH: 0799.60058
Digital Object Identifier: doi:10.1214/aoap/1177005359
Project Euclid: euclid.aoap/1177005359
[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.
Mathematical Reviews (MathSciNet): MR1181197
[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.
Mathematical Reviews (MathSciNet): MR1898854
Zentralblatt MATH: 1031.82032
Digital Object Identifier: doi:10.1023/A:1014755529138
[12] Funaki, T. (2005). Stochastic interface models. In Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1869 103–274. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR2228384
Zentralblatt MATH: 1119.60081
Digital Object Identifier: doi:10.1007/11429579_2
[13] Giacomin, G. (2007). Random Polymer Models. Imperial College Press, London.
Mathematical Reviews (MathSciNet): MR2380992
Zentralblatt MATH: 1125.82001
[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.
Mathematical Reviews (MathSciNet): MR2550363
Zentralblatt MATH: 1187.82076
Digital Object Identifier: doi:10.1007/s00440-008-0189-z
[16] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI.
Mathematical Reviews (MathSciNet): MR2466937
Zentralblatt MATH: 1160.60001
[17] Luby, M., Randall, D. and Sinclair, A. (2001). Markov chain algorithms for planar lattice structures. SIAM J. Comput. 31 167–192 (electronic).
Mathematical Reviews (MathSciNet): MR1857394
Zentralblatt MATH: 0992.82013
Digital Object Identifier: doi:10.1137/S0097539799360355
[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.
Mathematical Reviews (MathSciNet): MR1746301
Zentralblatt MATH: 1051.82514
Digital Object Identifier: doi:10.1007/978-3-540-48115-7_2
[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.
Mathematical Reviews (MathSciNet): MR1269387
Zentralblatt MATH: 0793.60110
Digital Object Identifier: doi:10.1007/BF02101929
Project Euclid: euclid.cmp/1104270006
[20] Martinelli, F., Sinclair, A. and Weitz, D. (2004). Glauber dynamics on trees: Boundary conditions and mixing time. Comm. Math. Phys. 250 301–334.
Mathematical Reviews (MathSciNet): MR2094519
Zentralblatt MATH: 1076.82010
Digital Object Identifier: doi:10.1007/s00220-004-1147-y
[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.
Mathematical Reviews (MathSciNet): MR2606632
Digital Object Identifier: doi:10.1007/s00220-009-0963-5
[22] Peres, Y. (2005). Mixing for Markov chains and spin systems. Lecture Notes for PIMS Summer School at UBC, August 2005. Available at www.stat.berkeley.edu/~peres/ubc.pdf.
[23] Posta, G. (1995/97). Spectral gap for an unrestricted Kawasaki type dynamics. ESAIM Probab. Stat. 1 145–181 (electronic).
Mathematical Reviews (MathSciNet): MR1444246
Digital Object Identifier: doi:10.1051/ps:1997106
[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.
Mathematical Reviews (MathSciNet): MR971048
Zentralblatt MATH: 1086.82503
Digital Object Identifier: doi:10.1007/BF01014902
[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.
Mathematical Reviews (MathSciNet): MR1016150
[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.
Mathematical Reviews (MathSciNet): MR1611693
[27] Randall, D. and Tetali, P. (2000). Analyzing Glauber dynamics by comparison of Markov chains. J. Math. Phys. 41 1598–1615.
Mathematical Reviews (MathSciNet): MR1757972
Zentralblatt MATH: 0974.60052
Digital Object Identifier: doi:10.1063/1.533199
[28] Spitzer, F. (1964). Principles of Random Walk. Van Nostrand, Princeton, NJ.
Mathematical Reviews (MathSciNet): MR171290
[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.
Mathematical Reviews (MathSciNet): MR1182416
Zentralblatt MATH: 0758.60070
Digital Object Identifier: doi:10.1007/BF02096629
Project Euclid: euclid.cmp/1104251144
[30] Wilson, D. B. (2004). Mixing times of Lozenge tiling and card shuffling Markov chains. Ann. Appl. Probab. 14 274–325.
Mathematical Reviews (MathSciNet): MR2023023
Zentralblatt MATH: 1040.60063
Digital Object Identifier: doi:10.1214/aoap/1075828054
Project Euclid: euclid.aoap/1075828054
The Annals of Applied Probability
