Mathematical Physics/Probability Theory
Cut-off and exit from metastability: two sides of the same coin
[Cut-off et sortie de la métastabilité : les deux faces de la même pièce]
Comptes Rendus. Mathématique, Tome 346 (2008) no. 11-12, pp. 691-696.

Nous présentons un cadre général qui relie cut-off et excursions de sortie pour des processus de naissance et de mort sur un alphabet dénombrable. Sous des hypothèses adaptées, nous montrons que le cut-off vers un équilibre (local) est accompagné par une distribution exponentielle des temps de sortie de l'équilibre. De plus, les trajectoires atypiques menant à ces excursions sont les renversées temporelles de trajectoires de cut-off ; en particulier leurs durées suivent la même loi.

We present a general framework linking cut-off and exit excursions for birth-and-death processes on a countable alphabet. Under suitable hypotheses, we prove that cut-off convergence towards a (local) equilibrium is accompanied by exponentially distributed out-of-equilibrium excursions. Furthermore, atypical trajectories leading to these excursions and final cut-off trajectories are related by time inversion; in particular their time lengths have identical laws.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2008.04.007
Bertoncini, Olivier 1 ; Barrera M., Javiera 2 ; Fernández, Roberto 1

1 Laboratoire de mathématiques Raphaël-Salem, UMR 6085 CNRS-Université de Rouen, avenue de l'Université, BP 12, 76801 Saint Étienne du Rouvray, France
2 Departamento de Matemática, Universidad Técnica Federico Sta. María, Av. España, 1680 Casilla 110-V, Valparaíso, Chile
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Bertoncini, Olivier; Barrera M., Javiera; Fernández, Roberto. Cut-off and exit from metastability: two sides of the same coin. Comptes Rendus. Mathématique, Tome 346 (2008) no. 11-12, pp. 691-696. doi : 10.1016/j.crma.2008.04.007. http://www.numdam.org/articles/10.1016/j.crma.2008.04.007/

[1] Aldous, D. Random walks on finite groups and rapidly mixing Markov chains, Seminar on Probability, XVII, Lecture Notes in Math., vol. 986, Springer, Berlin, 1983, pp. 243-297

[2] Aldous, D.; Diaconis, P. Shuffling cards and stopping times, Amer. Math. Monthly, Volume 93 (1986) no. 5, pp. 333-348

[3] Aldous, D.; Diaconis, P. Strong uniform times and finite random walks, Adv. Appl. Math., Volume 8 (1987) no. 1, pp. 69-97

[4] J. Barrera M., O. Bertoncini, R. Fernández, Abrupt convergence and metastability for birth and death chains, in preparation

[5] O. Bertoncini, Convergence abrupte et métastabilité, Thesis, Université de Rouen, 2007

[6] Bovier, A.; Eckhoff, M.; Gayrard, V.; Klein, M. Metastability in stochastic dynamics of disordered mean-field models, Probab. Theory Related Fields, Volume 119 (2001) no. 1, pp. 99-161

[7] Bovier, A.; Eckhoff, M.; Gayrard, V.; Klein, M. Metastability and low lying spectra in reversible Markov chains, Commun. Math. Phys., Volume 228 (2002) no. 2, pp. 219-255

[8] Cassandro, M.; Galves, A.; Olivieri, E.; Vares, M.E. Metastable behavior of stochastic dynamics: a pathwise approach, J. Statist. Phys., Volume 35 (1984) no. 5–6, pp. 603-634

[9] Diaconis, P. Group Representations in Probability and Statistics, Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 11, Institute of Mathematical Statistics, Hayward, CA, 1988

[10] Diaconis, P. The cutoff phenomenon in finite Markov chains, Proc. Natl. Acad. Sci. USA, Volume 93 (1996) no. 4, pp. 1659-1664

[11] Martínez, S.; Ycart, B. Decay rates and cutoff for convergence and hitting times of Markov chains with countably infinite state space, Adv. Appl. Probab., Volume 33 (2001) no. 1, pp. 188-205

[12] Olivieri, E.; Scoppola, E. Markov chains with exponentially small transition probabilities: first exit problem from a general domain. I. The reversible case, J. Statist. Phys., Volume 79 (1995) no. 3–4, pp. 613-647

[13] Olivieri, E.; Scoppola, E. Markov chains with exponentially small transition probabilities: first exit problem from a general domain. II. The general case, J. Statist. Phys., Volume 84 (1996) no. 5–6, pp. 987-1041

[14] Olivieri, E.; Vares, M.E. Large Deviations and Metastability, Encyclopedia of Mathematics and its Applications, vol. 100, Cambridge University Press, Cambridge, 2005

[15] Saloff-Coste, L. Lectures on finite Markov chains, Saint-Flour, 1996 (Lecture Notes in Math.), Volume vol. 1665, Springer, Berlin (1997), pp. 301-413

[16] Saloff-Coste, L. Random walks on finite groups, Probability on Discrete Structures, Encyclopaedia Math. Sci., vol. 110, Springer, Berlin, 2004, pp. 263-346

[17] Schonmann, R.H. The pattern of escape from metastability of a stochastic Ising model, Commun. Math. Phys., Volume 147 (1992) no. 2, pp. 231-240

[18] Scoppola, E. Metastability for Markov chains: a general procedure based on renormalization group ideas, Cambridge, 1993 (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.), Volume vol. 420, Kluwer Acad. Publ., Dordrecht (1994), pp. 303-322

[19] Ycart, B. Cutoff for Markov chains: some examples and applications, Santiago, 1998 (Nonlinear Phenom. Complex Systems), Volume vol. 6, Kluwer Acad. Publ., Dordrecht (2001), pp. 261-300

[20] Ycart, B. Modèles et algorithmes markoviens, Mathématiques & Applications (Berlin), vol. 39, Springer-Verlag, Berlin, 2002

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