Metastability in stochastic dynamics of disordered mean-field models

Probability Theory and Related Fields

Volumn 119, Issue 1, 2001, Pages 99-161

  Bovier, Anton ^a   Eckhoff, Michael ^b   Gayrard, Véronique ^c   Klein, Markus ^b  

^a WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

^b UNIVERSITY OF POTSDAM   (Germany)

^c CNRS   (France)

Author keywords

Disordered systems; Freidlin theory; Markov chains; Mean field models; Metastability; Random field Curie; Stochastic dynamics; Weiss model; Wentzell

Indexed keywords

EID: 0035605848     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/PL00012740     Document Type: Article

References (36)

1. Azencott, R.: "Petites perturbations aléatoires des systèmes dynamiques: développements asymptotiques", Bull. Sc. Math., 109, 253-308 (1985)
2. Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: "Metastability and low lying spectra in reversible Markov chains", WIAS-preprint 601 (2000)
3. Bovier, A., Gayrard, V.: "Hopfield models as generalized random mean field models", in "Mathematical aspects of spin glasses and neural networks", A. Bovier and P. Picco (eds.), Progress in Probability 41, (Birkhäuser, Boston 1998)
4. [BG2) Bovier, A., Gayrard, V.: "Sample path large deviations for a class of Markov chains related to disordered mean field models", WIAS-preprint 487 (1999)
5. Burke, C.J., Rosenblatt, M.: "A Markovian function of a Markov Chain", Ann. Math. Statist., 29, 1112-1122, (1958)
6. Catoni, O. Cerf, R.: "The exit path of a Markov chain with rare transitions", ESAIM Probab. Statist. 1, 95-144 (1995/97)
7. Cassandro, M., Galves, A., Olivieri, E., Varés, M.E.: "Metastable behaviour of stochastic dynamics: a pathwise approach, J. Stat. Phys. 35, 603-634 (1988)
8. Doyle, P.G., Snell, J.L.: "Random walks and electrical networks", Carus Mathematical Monographs, 22. Mathematical Association of America, Washington, DC, (1984)
9. Eckhoff, M., Klein, M.: Long-time behaviour of one-dimensional stochastic dynamics, preprint 456 SFB288, TU-Berlin (1999)
10. Fleming, W.H., James, M.R.: "Asymptotic series and exit time probabilities", Ann. Probab., 20, 1369-1384 (1992)
11. Fontes, R.E., Mathieu, P., Picco, P.: "On the averaged dynamics of the random field Curie-Weiss model", private communication
12. Freidlin, M.I., Wentzell, A.D.: "Random perturbations of dynamical systems", Springer, Berlin-Heidelberg-New York, (1984)
13. Gaudron, I., Trouvé, A.: "Fluctuations of empirical means at low temperature for finite Markov chains with rare transitions in the general case". Probab. Theor. Rel. Fields, 111, 215-251 (1998)
14. Dai Pra, P., den Hollander, F: "McKean-Vlasov limit for interacting random processes in random media", J. Statist. Phys., 84, 735-772 (1996)
15. Hopfield, J.J.: "Neural networks and physical systems with emergent collective computational abilities", Proc. Natl. Acad. Sci., USA 79, 2554-2558 (1982)
16. Helffer, B., Sjöstrand, J.: "Multiple wells in the semiclassical limit. III. Interaction through nonresonant wells", Math. Nachr. 124, 263-313 (1985)
17. van Kampen, N.G.: "Stochastic processes in physics and chemistry", North-Holland, Amsterdam, 1981 (reprinted in 1990)
18. Knessl, C., Matkowsky, B.J., Schuss, Z., Tier, C.: "An asymptotic theory of large deviations for Markov jump processes", SIAM J. Appl. Math., 46, 1006-1028 (1985)
19. Kifer, Y.: "On the principal eigenvalue of a singular perturbation problem with hyperbolic limit points and circles", J. Diff. Eq., 37, 108-139 (1980)
20. Külske, Ch.: "Metastates in disordered mean-field models: random field and Hopfield models", J. Statist. Phys., 88, 1257-1293 (1997)
21. Kifer, Y.: "The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point", Israel J. Math., 40, 74-94 (1981)
22. Kifer, Y: "Random perturbations of dynamical systems", Progress in Probability and Statistics 16, Birkhäuser, Boston-Basel, (1988)
23. Kifer, Y: "A discrete time version of the Wentzell-Freidlin theory", Ann. Probab. 18, 1676-1692 (1990)
24. Kurtz, T.G.: "Solutions of ordinary differential equations as limits of pure jump Markov processes", J. Appl. Probab., 7, 49-58 (1970)
25. Liggett, T.M.: "Interacting particle systems", Springer, Berlin, (1985)
26. Mathieu, P., Picco, P.: "Metastability and convergence to equilibrium for the random field Curie-Weiss model", J. Stat. Phys., 91, 679-732 (1998)
27. Newman, C.M., Stein, D.L.: "Thermodynamic chaos and the structure of short-range spin glasses", in Mathematical aspects of spin glasses and neural networks, A. Bovier and P. Picco (eds.), Progress in Probability 41, Birkhäuser, Boston-Basel-Berlin, (1998)
28. Olivieri, E., Scoppola, E.: "Markov chains with exponentially small transition probabilities: First exit problem from a general domain -I. The reversible case" J. Stat. Phys., 79, 613 (1995)
29. Olivieri, E., Scoppola, E.: "Markov chains with exponentially small transition probabilities: First exit problem from a general domain-II. The general case". J. Stat. Phys., 84, 987-1041 (1996)
30. Schonmann, R., Shlosman, S.: "Wulff droplets and the metastable relaxation of kinetic Ising models", Commun. Math. Phys., 194, 389-462 (1998)
31. Scoppola, E.: "Renormalization group for Markov chains and application to meta" stability", J. Statist. Phys. 73, 83-121 (1993)
32. Vares, M.E.: "Large deviations and metastability", in Disordered systems, 1-62, Travaux en Cours Vol. 53, Hermann, Paris, (1996)
33. Wentzell, A.D.: "Rough limit theorems on large deviations for Markov stochastic processes I.", Theor. Probab. Appl., 21, 227-242 (1976)
34. Wentzell, A.D.: "Rough limit theorems on large deviations for Markov stochastic processes II.", Theor. Probab. Appl., 21, 499-512 (1976)
35. Wentzell, A.D.: "Rough limit theorems on large deviations for Markov stochastic processes III.", Theor. Probab. Appl., 24, 675-692 (1979)
36. Wentzell, A.D.: "Rough limit theorems on large deviations for Markov stochastic processes IV.", Theor. Probab. Appl., 27, 215-234 (1982)