Abstract.
We study continuous time Glauber dynamics for random configurations with local constraints (e.g. proper coloring, Ising and Potts models) on finite graphs with n vertices and of bounded degree. We show that the relaxation time (defined as the reciprocal of the spectral gap |λ1−λ2|) for the dynamics on trees and on planar hyperbolic graphs, is polynomial in n. For these hyperbolic graphs, this yields a general polynomial sampling algorithm for random configurations. We then show that for general graphs, if the relaxation time τ2 satisfies τ2=O(1), then the correlation coefficient, and the mutual information, between any local function (which depends only on the configuration in a fixed window) and the boundary conditions, decays exponentially in the distance between the window and the boundary. For the Ising model on a regular tree, this condition is sharp.
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Aldous, D., Fill, J.A.: Reversible Markov chains and random walks on graphs. Book in preparation 2000, Current version available at http://www.stat.berkeley.edu/users/aldous/book.html
van den Berg, J.: A uniqueness condition for Gibbs measures, with application to the 2-dimensional Ising antiferromagnet. Comm. Math. Phys. 152 (1), 161–166 (1993)
Bleher, P.M., Ruiz, J., Zagrebnov, V.A.: On the purity of limiting Gibbs state for the Ising model on the Bethe lattice. J. Stat. Phys 79, 473–482 (1995)
Bubley, R., Dyer, M.: Path coupling: a technique for proving rapid mixing in Markov chains. In: Proceedings of the 38th Annual Symposium on Foundations of Computer Science (FOCS), 1997, pp. 223–231
Chen, M.F.: Trilogy of couplings and general formulas for lower bound of spectral gap. Probability towards 2000, Lecture Notes in Statist., 128, Springer, New York, 1998, pp. 123–136
Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley, New York, 1991
Dyer, M., Greenhill, C.: On Markov chains for independent sets. J. Algor. 35, 17–49 (2000)
Evans, W., Kenyon, C., Peres, Y., Schulman, L.J.: Broadcasting on trees and the Ising Model. Ann. Appl. Prob. 10, 410–433 (2000)
Fortuin, C.M., Kasteleyn, P.W.: On the random-cluster model. I. Introduction and relation to other models. Physica 57, 536–564 (1972)
Fortuin, C.M., Kasteleyn, P.W., Ginibre, J.: Correlation inequalities on some partially ordered sets. Comm. Math. Phys. 22, 89–103 (1971)
Haggstrom, O., Jonasson, J., Lyons, R.: Explicit isoperimetric constants and phase transitions in the random-cluster model. Ann. Probab. 30, 443–473 (2002)
Ioffe, D.: A note on the extremality of the disordered state for the Ising model on the Bethe lattice. Lett. Math. Phys. 37, 137–143 (1996)
Janson, S., Luczak, T., Ruciński, A.: Random Graphs. Wiley, New York, 2000
Jerrum, M.: A very simple algorithm for estimating the number of k-colorings of a low-degree graph. Rand. Struc. Alg. 7, 157–165 (1995)
Jerrum, M., Sinclair, A.: Approximating the permanent. Siam J. Comput. 18, 1149–1178 (1989)
Jerrum, M., Sinclair, A.: Polynomial time approximation algorithms for the Ising model. Siam J. Comput. 22, 1087–1116 (1993)
Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries. Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, Crete, Greece, 2001
Katok, S.: Fuchsian Groups. University of Chicago Press, 1992
Kenyon, C., Mossel, E., Peres, Y.: Glauber dynamics on trees and hyperbolic graphs. 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas, NV, 2001), IEEE Computer Soc., Los Alamitos, CA, 2001, pp. 568–578
Kinnersley, N.G.: The vertex seperation number of a graph equals its path-width. Infor. Proc. Lett. 42, 345–350 (1992)
Liggett, T.: Interacting particle systems. Springer, New York, 1985
Luby, M., Vigoda, E.: Approximately Counting Up To Four. In: proceedings of the 29th Annual Symposium on Theory of Computing (STOC), 1997, pp. 682–687
Luby, M., Vigoda, E.: Fast Convergence of the Glauber Dynamics for Sampling Independent Sets, Statistical physics methods in discrete probability, combinatorics and theoretical computer science. Rand. Struc. Alg. 15, 229–241 (1999)
Magnus, W.: Noneuclidean tessellations and their groups. Academic Press, New York and London, 1974
Martinelli, F.: Lectures on Glauber dynamics for discrete spin models. Lectures on probability theory and statistics (Saint-Flour, 1997) Lecture Notes in Math. 1717, Springer, Berlin, 1998, pp. 93–191
Martinelli, F., Sinclair, A., Weitz, D.: Glauber dynamics on trees: Boundary conditions and mixing time. Preprint 2003, available at http://front.math.ucdavis.edu/math.PR/0307336
Mossel, E.: Reconstruction on trees: Beating the second eigenvalue. Ann. Appl. Probab. 11 (1), 285–300 (2001)
Mossel, E.: Recursive reconstruction on periodic trees. Rand. Struc. Alg. 13, 81–97 (1998)
Mossel, E., Peres Y.: Information flow on trees. To appear in Ann. Appl. Probab., 2003
Nacu, S.: Glauber dynamics on the cycle is monotone. To appear, Probab. Theory Related Fields, 2003
Paterson, A.L.T.: Amenability. American Mathematical Soc., Providence, 1988
Propp, J., Wilson, D.: Exact Sampling with Coupled Markov Chains and Applications to Statistical Mechanics. Rand. Struc. Alg. 9, 223–252 (1996)
Peres, Y., Winkler, P.: In preparation, 2003
Randall, D., Tetali, P.: Analyzing Glauber dynamics by comparison of Markov chains. J. Math. Phys. 41, 1598–1615 (2000)
Robertson, N., Seymour, P.D.: Graph minors. I. Excluding a forest. J. Comb. Theory Series B 35, 39–61 (1983)
Saloff-Coste, L.: Lectures on finite Markov chains. Lectures on probability theory and statistics (Saint-Flour, 1996) Lecture Notes in Math. 1665, Springer, Berlin, 1997, pp. 301–413
Vigoda, E.: Improved bounds for sampling colorings. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. J. Math. Phys. 41 (3), 1555–1569 (2001)
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Research supported by Microsoft graduate fellowship.
Supported by a visiting position at INRIA and a PostDoc at Microsoft research.
Research supported by NSF Grants DMS-0104073, CCR-0121555 and a Miller Professorship at UC Berkeley.
Acknowledgement We are grateful to David Aldous, David Levin, Laurent Saloff-Coste and Peter Winkler for useful discussions. We thank Dror Weitz for helpful comments on [19].
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Berger, N., Kenyon, C., Mossel, E. et al. Glauber dynamics on trees and hyperbolic graphs. Probab. Theory Relat. Fields 131, 311–340 (2005). https://doi.org/10.1007/s00440-004-0369-4
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DOI: https://doi.org/10.1007/s00440-004-0369-4