The Annals of Probability

Pointwise estimates and exponential laws in metastable systems via coupling methods

Alessandra Bianchi, Anton Bovier, and Dmitry Ioffe

Full-text: Open access

Abstract

We show how coupling techniques can be used in some metastable systems to prove that mean metastable exit times are almost constant as functions of the starting microscopic configuration within a “meta-stable set.” In the example of the Random Field Curie Weiss model, we show that these ideas can also be used to prove asymptotic exponentiallity of normalized metastable escape times.

Article information

Source
Ann. Probab., Volume 40, Number 1 (2012), 339-371.

Dates
First available in Project Euclid: 3 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1325605005

Digital Object Identifier
doi:10.1214/10-AOP622

Mathematical Reviews number (MathSciNet)
MR2917775

Zentralblatt MATH identifier
1237.82039

Subjects
Primary: 82C44: Dynamics of disordered systems (random Ising systems, etc.) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60G70: Extreme value theory; extremal processes

Keywords
Disordered system random field Curie–Weiss model Glauber dynamics metastability potential theory coupling exponential law

Citation

Bianchi, Alessandra; Bovier, Anton; Ioffe, Dmitry. Pointwise estimates and exponential laws in metastable systems via coupling methods. Ann. Probab. 40 (2012), no. 1, 339--371. doi:10.1214/10-AOP622. https://projecteuclid.org/euclid.aop/1325605005


Export citation

References

  • [1] Bianchi, A., Bovier, A. and Ioffe, D. (2009). Sharp asymptotics for metastability in the random field Curie–Weiss model. Electron. J. Probab. 14 1541–1603.
  • [2] Bovier, A. (2009). Metastability. In Methods of Contemporary Mathematical Statistical Physics. Lecture Notes in Math. 1970 177–221. Springer, Berlin.
  • [3] Bovier, A., den Hollander, F. and Spitoni, C. (2010). Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes and low temperature. Ann. Probab. 38 661–713.
  • [4] Bovier, A., Eckhoff, M., Gayrard, V. and Klein, M. (2001). Metastability in stochastic dynamics of disordered mean-field models. Probab. Theory Related Fields 119 99–161.
  • [5] Bovier, A., Eckhoff, M., Gayrard, V. and Klein, M. (2002). Metastability and low lying spectra in reversible Markov chains. Comm. Math. Phys. 228 219–255.
  • [6] Bovier, A., Eckhoff, M., Gayrard, V. and Klein, M. (2004). Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times. J. Eur. Math. Soc. (JEMS) 6 399–424.
  • [7] Cassandro, M., Galves, A., Olivieri, E. and Vares, M. E. (1984). Metastable behavior of stochastic dynamics: A pathwise approach. J. Stat. Phys. 35 603–634.
  • [8] Freidlin, M. I. and Wentzell, A. D. (1984). Random Perturbations of Dynamical Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 260. Springer, New York.
  • [9] Levin, D. A., Luczak, M. J. and Peres, Y. (2010). Glauber dynamics for the mean-field Ising model: Cut-off, critical power law, and metastability. Probab. Theory Related Fields 146 223–265.
  • [10] Martinelli, F., Olivieri, E. and Scoppola, E. (1989). Small random perturbations of finite- and infinite-dimensional dynamical systems: Unpredictability of exit times. J. Stat. Phys. 55 477–504.
  • [11] Martinelli, F. and Scoppola, E. (1988). Small random perturbations of dynamical systems: Exponential loss of memory of the initial condition. Comm. Math. Phys. 120 25–69.
  • [12] Mathieu, P. and Picco, P. (1998). Metastability and convergence to equilibrium for the random field Curie–Weiss model. J. Stat. Phys. 91 679–732.
  • [13] Nummelin, E. (1978). A splitting technique for Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete 43 309–318.
  • [14] Olivieri, E. and Vares, M. E. (2005). Large Deviations and Metastability. Encyclopedia of Mathematics and Its Applications 100. Cambridge Univ. Press, Cambridge.