Counting without sampling: Asymptotics of the log-partition function for certain statistical physics models

Random Structures and Algorithms

Volumn 33, Issue 4, 2008, Pages 452-479

  Bandyopadhyay, Antar ^a   Gamarnik, David ^b  

^a INDIAN STATISTICAL INSTITUTE   (India)

^b MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Approximation algorithms; Correlation decay; Gibbs measures; Graph counting

Indexed keywords

EID: 60349088691     PISSN: 10429832     EISSN: 10982418     Source Type: Journal    
DOI: 10.1002/rsa.20236     Document Type: Conference Paper

References (42)

1. D. Aldous and A. Bandyopadhyay, A survey of max-type recursive distributional equations, Ann Appl Probab 15 (2005), 1047-1110.
2. D. Aldous, The ζ (2) limit in the random assignment problem. Random Struct Algorithms (2001), 381-418.
3. D. Achlioptas and C. Moore, The chromatic number of random regular graphs, 8th edition, Workshop on Randomization and Computation (RANDOM), 2004.
4. D. Aldous and J. M. Steele, The objective method: Probabilistic combinatorial optimization and local weak convergence, In: H. Kesten, editor, Discrete combinatorial probability, Springer-Verlag, New York, 2003, pp. 1-72.
5. A. Bandyopadhyay, Hard-core model on random graphs, in preparation.
6. A. Bandyopadhyay, Bivariate uniqueness in the logistic fixed point equation, Technical Report 629, Department of Statistics, UC, Berkeley, 2002.
7. I. Bezakova, D. Stefankovic, V. Vazirani, and E. Vigoda, Accelerated simulated annealing algorithm for the permanent and combinatorial counting problems, SLAM J Comput 37 (2008), 1429-1454.
8. G. Brightwell and P. Winkler, Random colorings of a Cayley tree, In B. Bollobas, editor, Contemporary Combinatorics, Bolyai Society Mathematical Studies, 2002, pp. 247-276.
9. G. Brightwell and P. Winkler, Graph homomorphisms and long range action, In Nesetril and Winkler, editors, Graphs, morphisms and statistical physics, DIMACS series in discrete mathematics and computer science, 2004, pp. 29-47.
10. G. Brightwell and P. Winkler, A second threshold for the hard-core model on a Bethe lattice, Random Struct Algorithms 24 (2004), 303-314.
11. M. E. Dyer, A. Frieze, and R. Kannan, A random polynomial time algorithm for approximating the volume of convex bodies, I Assoc Comput Machinery 38 (1991), 1-17.
12. M. Dyer, A. Frieze, T. Hayes, and E. Vigoda, Randomly coloring constant degree graphs, In Proceedings of 45th IEEE Symposium on Foundations of Computer Science, Rome, Italy, October 2004.
13. M. Dyer, L. A. Goldberg, and M. Jerrum, Counting and sampling H-colourings, Inform Comput 189 (2004), 1-16.
14. R. L. Dobrushin, Prescribing a system of random variables by the help of conditional distributions, Theory Probab Appl 15 (1970), 469-497.
15. D. Gamarnik, Linear phase transition in random linear constraint satisfaction problems, Probab Theory Related Fields 129 (2004), 410-440.
16. H. O. Georgii, Gibbs measures and phase transitions, de Gruyter studies in mathematics 9, Walter de Gruyter & Co., Berlin, 1988.
17. D. Gamarnik, T. Nowicki, and G. Swirszcz, Invariant probability measures and dynamics of exponential linear type maps, Ergodic Theory and Dynamical Systems, to appear.
18. D. Gamarnik, T. Nowicki, and G. Swirscsz, Maximum weight independent sets and matchings in sparse random graphs. Exact results using the local weak convergence method, Random Struct Algorithms 28 (2006), 76-106.
19. S. lanson, T. Łuczak, and A. Rucinski, Random graphs, Wiley, New York, 2000.
20. J. Jonasson, Uniqueness of uniform random colorings of regular trees, Statistics Probab Lett 57 (2002), 243-248.
21. M. Jerrum and A. Sinclair, Approximating the permanent, SLAM J Comput 18 (1989), 1149-1178.
22. M. Jerrum and A. Sinclair, The Markov chain Monte Carlo method: An approach to approximate counting and integration, In: D. Hochhaum, editor, Approximation algorithms for NP-hard problems, PWS Publishing Company, Boston, MA, 1997.
23. M. Jerrum, A. Sinclair, and E. Vigoda, A polynomial-time approximation algorithms for permanent of a matrix with non-negative entries, J Assoc Comput Mach 51 (2004), 671-697.
24. F. Kelly, Stochastic models of computer communication systems, J R Statist Soc B 47 (1985), 379-395.
25. 5) volume algorithm for convex bodies, Random Struct Algorithms 11 (1997), 1-50.
26. M. Luby and E. Vigoda, Approximately counting up to four, Proc 29d Ann ACM Symposium on the Theory of Computing (STOC), El Paso, Texas, May 1997.
27. 4) volume algorithm, Proceedings of the 44th annual IEEE Symposium on Foundations of Computer Science, Cambridge, MA, October 2003, pp. 650-659.
28. J. Martin, Reconstruction thresholds on regular trees, In: C. Banderier and C. Krattenthaler, editors, Discrete Random Walks, DRW 03, Discrete Mathematics and Theoretical Computer Science Proceedings AC, 2003, pp. 191-2003.
29. M. Mezard and A. Montanari, Reconstruction on trees and spin glass transition, J Stat Phys 124 (2006).
30. E. Mossel, Survey: information flow on trees, In J. Nestril and P. Winkler, editors, Graphs, morphisms and statistical physiscs, DIMACS series in discrete mathematics and theoretical computer science, American Mathematical Society, 2004, pp. 155-170.
31. E. Mossel and Y. Peres, Information flow on trees, Ann Appl Probab 13 (2003), 817-844.
32. M. Mezard and G. Parisi, The cavity method at zero temperature, http://fr.arxiv.org/ps/condmat/0207121, 2005.
33. M. Mezard, G. Parisi, and M. A. Virasoro, Spin-glass theory and beyond, Vol. 9: Lecture Notes in Physics, World Scientific, Singapore, 1987.
34. O. Rivoire, G. Biroli, O. C. Martin, and M. Mezard, Glass models on Bethe lattices, Eur Phys J B 37 (2004), 55-78.
35. F. Spitzer, Markov random fields on an infinite tree, Ann Probab 3 (1975), 387-398.
36. M. Talagrand, The high temperature case of the K-sat problem, Probab Theory Related Fields 119 (2001), 187-212.
37. M. Talagrand, Parisi formula, Ann Math 163 (2006), 221-263.
38. L. G. Valiant, The complexity of computing the permanent, Theor Comput Sci 8 (1979), 189-201.
39. J. Warren, Dynamics and endogeny for recursive processes on trees, http://arxiv.org/abs/math. PR/0506038, 2005.
40. D. Weitz, Combinatorial criteria for uniqueness of Gibbs measures, Random Struct Algorithms 27 (2005), 445-475.
41. D. Weitz, Counting independent sets up to the tree threshold. In Proc 38th Ann. Symposium on the Theory of Computing, Seattle, WA, May 2006.
42. S. Zachary, Countable state space Markov random fields and Markov chains on tree, Ann Prob 11 (1983), 894-903.