Exact sampling with coupled Markov chains and applications to statistical mechanics

Random Structures and Algorithms

Volumn 9, Issue 1, 1996, Pages 223-252

  Propp, James Gary ^a   Wilson, David Bruce ^a  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0005193926     PISSN: 10429832     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1098-2418(199608/09)9:1/2<223::AID-RSA14>3.0.CO;2-O     Document Type: Article

References (64)

1. D. J. Aldous, A random walk construction of uniform spanning trees and uniform labelled trees, SIAM J. Discrete Math., 3(4), 450-465 (1990).
2. D. Aldous, On simulating a Markov chain stationary distribution when transition probabilities are unknown, preprint, 1994.
3. D. Aldous and P. Diaconis, Strong uniform times and finite random walks, Adv. Appl. Math., 8(1), 69-97 (1987).
4. D. J. Aldous and J. A. Fill, Reversible Markov Chains and Random Walks on Graphs, to appear.
5. D. Aldous, L. Lovász, and P. Winkler, Fast mixing in a Markov chain, preprint, 1995.
6. N. Alon and J. H. Spencer, The Probabilistic Method, Wiley, New York, 1992, with appendix by Paul Erdos.
7. S. Asmussen, P. W. Glynn, and H. Thorisson, Stationary detection in the initial transient problem, ACM Trans. Model. Comput. Simul., 2(2), 130-157 (1992).
8. R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic, New York, 1982.
9. J. Besag, On the statistical analysis of dirty pictures, J. Roy. Stat. Soc. B, 48(3), 259-302 (1986).
10. J. Besag and P. Clifford, Generalized Monte Carlo significance tests, Biometrika, 76(4), 633-642 (1989).
11. N. Biggs, Algebraic Graph Theory, 2nd ed., Cambridge University Press, Cambridge, 1993.
12. J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. J. Newman, The Theory of Critical Phenomena: An Introduction to the Renormalization Group, Oxford University Press, Oxford, 1992.
13. A. Björner, Orderings of Coxeter groups, in Combinatorics and Algebra, Contemporary Mathematics. #34, American Mathematical Society, Providence, RI, 1984, pp. 175-195.
14. H. W. J. Blöte and H. J. Hilhorst, Roughening transitions and the zero-temperature triangular Ising antiferromagnet, J. Phys. A, 15(11), L631-L637 (1982).
15. A. Broder, Generating random spanning trees, Found. Comput. Sci., 442-447 (1989).
16. H. Cohn, N. Elkies, and J. Propp, Local statistics for random domino tilings of the Aztec diamond, Duke Math. J. (1996), to appear.
17. J. Conway and J. Lagarias, Tiling with polyominoes and combinatorial group theory, J. Combinat. Theor. Ser. A, 53, 183-208 (1990).
18. P. Diaconis, Group Representations in Probability and Statistics, Institute of Mathematical Statistics, Hayward, CA, 1988.
19. P. Diaconis and L. Saloff-Coste, What do we know about the Metropolis algorithm?, ACM Symp. Theory of Computing, 1995, pp. 112-129.
20. P. Diaconis and D. Stroock, Geometric bounds for eigenvalues of Markov chains, Ann. Appl. Probab. 1(1), 36-61 (1991).
21. P. Diaconis and B. Sturmfels, Algebraic algorithms for sampling from conditional distributions, Ann. Stat., to appear.
22. S. Felsner and L. Wernisch, Markov chains for linear extensions, the two-dimensional case, preprint, 1996.
23. C. M. Fortuin and P. W. Kasteleyn, On the random cluster model. I. Introduction and relation to other models, Physica, 57(4), 536-564 (1972).
24. C. M. Fortuin, P. W. Kasteleyn, and J. Ginibre, Correlation inequalities on some partially ordered sets, Commun. Math. Phys., 22, 89-103 (1971).
25. A. Frigessi, C.-R. Hwang, S.-J. Sheu, and P. di Stefano, Convergence rates of the Gibbs sampler, the Metropolis algorithm and other single-site updating dynamics, J. Roy. Stat. Soc. B, 55(1), 205-219 (1993).
26. S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Machine Intell., PAMI-6(6), 721-724 (1984).
27. D. Griffeath, Additive and Cancellative Interacting Particle Systems, Lecture Notes in Mathematics 724, Springer-Verlag, New York, 1979.
28. G. Grimmett, Percolation, Springer-Verlag, New York, 1989.
29. G. Grimmett, The stochastic random-cluster process and the uniqueness of random-cluster measures, Ann. Prob., 23(4), 1461-1510 (1995).
30. M. R. Henzinger and V. King, Randomized dynamic graph algorithms with polylogarithmic time per operation, ACM Symp. Theory of Computing, 1995, pp. 519-527.
31. R. Holley, Remarks on the FKG inequalities, Commun. Math. Phys., 36, 227-231 (1974).
32. R. Holley and T. Liggett, Ergodic theorems for weakly interacting systems and the voter model, Ann. Probab., 3, 643-663 (1975).
33. S. Ingrassia, On the rate of convergence of the Metropolis algorithm and Gibbs sampler by geometric bounds, Ann. Appl. Probab., 4(2), 347-389 (1994).
34. M. Jerrum and A. Sinclair, Approximating the permanent, SIAM J. Comput., 18(6), 1149-1178 (1989).
35. W. Jockusch, J. Propp, and P. Shor, Random domino tilings and the arctic circle theorem, preprint, 1995.
36. V. E. Johnson, Testing for convergence of Markov chain Monte Carlo algorithms using parallel sample paths, preprint, 1995.
37. M. G. Kendall, Rank Correlation Methods, 3rd ed., Hafner, New York, 1962.
38. J. Kim, P. Shor, and P. Winkler, private communication.
39. L. S. Levitov, Equivalence of the dimer resonating-valence-bond problem to the quantum roughening problem, Phys. Rev. Lett., 64(1), 92-94 (1990).
40. E. Lieb, Residual entropy of square ice, Phys. Rev., 162, 162-172 (1967).
41. T. Liggett, Interacting Particle Systems, Springer-Verlag, New York, 1985.
42. T. Lindvall, Lectures on the Coupling Method, Wiley, New York, 1992.
43. L. Lovász and M. Simonovits, On the randomized complexity of volume and diameter, Found. Comput. Sci., 482-491 (1992).
44. L. Lovász and P. Winkler, Exact mixing in an unknown Markov chain, Electronic J. Combinat., 2 (1995) Paper #R15.
45. M. Luby, D. Randall, and A. Sinclair, Markov chain algorithms for planar lattice structures (extended abstract), Found. Comput. Sci., 150-159 (1995).
46. F. Martinelli, E. Olivieri, and R. H. Schonmann, For 2-d lattice spin systems weak mixing implies strong mixing, Commun. Math. Phys., 165(1), 33-47 (1994).
47. J. Propp, Lattice structure for orientations of graphs, preprint, 1993.
48. J. S. Rosenthal, Minorization conditions and convergence rates for Markov chain Monte Carlo, J. Am. Stat. Assoc., 90(430), 558-566 (1995).
49. R. H. Schonmann, Slow droplet-driven relaxation of stochastic Ising models in the vicinity of the phase coexistence region, Commun. Math. Phys., 161(1), 1-49 (1994).
50. A. Sinclair, Algorithms for Random Generation and Counting: A Markov Chain Approach, Birkhäuser, Boston, 1993.
51. A. D. Sokal, Monte Carlo methods in statistical mechanics: foundations and new algorithms, lecture notes from Cours de Troisième Cycle de la Physique en Suisse Romande, Lausanne, 1989.
52. R. P. Stanley, Enumerative Combinatorics, Wadsworth, Monterey, CA, 1986, Vol. 1.
53. D. W. Stroock and B. Zegarlinski, The logarithmic Sobolev inequality for discrete spin systems on a lattice, Commun. Math. Phys., 149(1), 175-193 (1992).
54. M. Sweeny, Monte Carlo study of weighted percolation clusters relevant to the Potts models, Phys. Rev. B, 27(7), 4445-4455 (1983).
55. R. H. Swendsen and J.-S. Wang, Nonuniversal critical dynamics in Monte Carlo simulations, Phys. Rev. Lett., 58(2), 86-88 (1987).
56. L. E. Thomas, Bound on the mass gap for finite volume stochastic Ising models at low temperature, Commun. Math. Phys., 126(1), 1-11 (1989).
57. W. Thurston, Conway's tiling groups, Am. Math. Mon., 97, 757-773 (1990).
58. H. van Beijeren, Exactly solvable model for the roughening transition of a crystal surface, Phys. Rev. Lett., 38(18), 993-996 (1977).
59. D. B. Wilson, work in progress.
60. D. B. Wilson, Generating random spanning trees more quickly than the cover time, ACM Symp. Theory of Computing, 1996, pp. 296-303.
61. D. B. Wilson and J. G. Propp, How to get a perfectly random sample from a generic Markov chain and generate a random spanning tree of a directed graph, ACM-SIAM Symp. Discrete Algorithms, 1996, pp. 448-457.
62. U. Wolff, Collective Monte Carlo updating for spin systems, Phys. Rev. Lett., 62(4), 361-364 (1989).
63. F. Y. Wu, The Potts model, Rev. Mod. Phys., 54(1), 235-268 (1982).
64. W. Zheng and S. Sachdev, Sine-Gordon theory of the non-Néel phase of two-dimensional quantum antiferromagnets, Phys. Rev. B, 40, 2704-2707 (1989).