The Ising model on trees: Boundary conditions and mixing time

Annual Symposium on Foundations of Computer Science - Proceedings

Volumn , Issue , 2003, Pages 628-639

  Martinelli, Fabio ^a   Sinclair, Alistair ^b   Weitz, Dror ^b  

^a ROMA TRE UNIVERSITY   (Italy)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY CONDITIONS; MATHEMATICAL MODELS; POLES AND ZEROS; POLYNOMIAL APPROXIMATION; THEOREM PROVING; TIME DOMAIN ANALYSIS; TREES (MATHEMATICS);

GLAUBER DYNAMICS; ISING MODEL; LOG SOBOLEV CONSTANT; POTTS MODEL;

COMBINATORIAL MATHEMATICS;

EID: 0344550472     PISSN: 02725428     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper

References (24)

1. D. Aldous, "Random walks on finite groups and rapidly mixing Markov chains," Séminaire de Probabilites XVII, Lecture Notes in Mathematics 986, pp. 243-297, Springer, Berlin, 1983.
2. R.J. Baxter, Exactly solved models in statistical mechanics, Academic Press, London, 1982.
3. N. Berger, C. Kenyon, E. Mossel and Y. Peres, "Glauber dynamics on trees and hyperbolic graphs," preprint, 2003.
4. Preliminary version: C. Kenyon, E. Mossel and Y. Peres, "Glauber dynamics on trees and hyperbolic graphs," Proc. 42nd IEEE Symposium on Foundations of Computer Science (2001), pp. 568-578.
5. L. Bertini, N. Cancrini and F. Cesi, "The spectral gap for a Glauber-type dynamics in a continuous gas," Ann. Inst. H. Poincaré Probab. Statist. 38 (2002), pp. 91-108.
6. P. Bleher, J. Ruiz, R. H. Schonmann, S. Shlosman and V. Zagrebvnov, "Rigidity of the critical phases on a Cayley tree," Moscow Math. J. 1 (2001), pp. 345-363.
7. P. Bleher, J. Ruiz and V. Zagrebnov, "On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice," J. Stat. Phys. 79 (1995), pp. 473-482.
8. T. Bodineau and F. Martinelli, "Some new results on the kinetic Ising model in a pure phase," J. Stat. Phys. 109 (1), 2002.
9. F. Cesi, "Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields," Probab. Theory Related Fields 120 (2001), pp. 569-584.
10. W. Evans, C. Kenyon, Y. Peres and L.J. Schulman, "Broadcasting on trees and the Ising model," Ann. Appl. Probab. 10 (2000), pp. 410-433.
11. D. Fisher and D. Huse, "Dynamics of droplet fluctuations in pure and random Ising systems," Physics Review B 35 (13), 1987.
12. H.-O. Georgii, Gibbs measures and phase transitions, de Gruyter Studies in Mathematics 9, Walter de Gruyter & Co., Berlin, 1988.
13. D. Ioffe, "A note on the extremality of the disordered state for the Ising model on the Bethe lattice," Letters in Mathematical Physics 37 (1996), pp. 137-143.
14. J. Jonasson and J.E. Steif, "Amenability and phase transition in the Ising model," J. Theoret. Probab. 12 (1999), pp. 549-559.
15. R. Lyons, "Phase transitions on non amenable graphs," J. Math. Phys. 41 (2000), pp. 1099-1127.
16. F. Martinelli, "Lectures on Glauber dynamics for discrete spin models," Lectures on Probability Theory and Statistics (Saint-Flour, 1997), Lecture notes in Mathematics 1717, pp. 93-191, Springer, Berlin, 1998.
17. F. Martinelli and E. Olivieri, "Approach to equilibrium of Glauber dynamics in the one phase region I: The attractive case," Comm. Math. Phys. 161 (1994), pp. 447-486.
18. F. Martinelli, E. Olivieri and R. Schonmann, "For 2-D lattice spin systems weak mixing implies strong mixing," Comm. Math. Phys. 165 (1994), pp. 33-47.
19. F. Martinelli, A. Sinclair and D. Weitz, "The Ising model on trees: Boundary conditions and mixing time," Tech. Report UCB//CSD-03-1256, Dept. of EECS, UC Berkeley, July 2003. http://sunsite.Berkeley.EDU/NCSTRL/
20. F. Martinelli, A. Sinclair and D. Weitz, "Fast mixing for independent sets, colorings and other models on trees," preprint, 2003.
21. Y. Peres and P. Winkler, personal communication.
22. C.J. Preston, Gibbs states on countable sets, Cambridge Tracts in Mathematics 68, Cambridge University Press, London, 1974.
23. L. Saloff-Coste, "Lectures on finite Markov chains," Lectures on probability theory and statistics (Saint-Flour, 1996), Lecture notes in Mathematics 1665, pp. 301-413, Springer, Berlin, 1997.
24. D.W. Stroock and B. Zegarlinski, "The logarithmic Sobolev inequality for discrete spin systems on a lattice," Comm. Math. Phys. 149 (1992), pp. 175-194.