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Journal of Statistical Physics

, Volume 160, Issue 6, pp 1449–1482 | Cite as

Metastability of Reversible Random Walks in Potential Fields

  • C. Landim
  • R. Misturini
  • K. Tsunoda
Article

Abstract

Let \(\Xi \) be an open and bounded subset of \({\mathbb {R}}^d\), and let \(F:\Xi \rightarrow {\mathbb {R}}\) be a twice continuously differentiable function. Denote by \(\Xi _N\) the discretization of \(\Xi \), \(\Xi _N = \Xi \cap (N^{-1} {\mathbb {Z}}^d)\), and denote by \(X_N(t)\) the continuous-time, nearest-neighbor, random walk on \(\Xi _N\) which jumps from \({\varvec{x}}\) to \({\varvec{y}}\) at rate \( e^{-(1/2) N [F({\varvec{y}}) - F({\varvec{x}})]}\). We examine in this article the metastable behavior of \(X_N(t)\) among the wells of the potential F.

Keywords

Reversible random walks Metastability Exit points 

Notes

Acknowledgments

R. Misturini was supported by CAPES and CNPq-Brazil during the preparation of this work, and K. Tsunoda was supported by the Program for Leading Graduate Course for Frontiers of Mathematical Sciences and Physics.

References

  1. 1.
    Beltrán, J., Landim, C.: Tunneling and metastability of continuous time Markov chains. J. Stat. Phys. 140, 1065–1114 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Beltrán, J., Landim, C.: Tunneling and metastability of continuous time Markov chains II. J. Stat. Phys. 149, 598–618 (2012)MathSciNetADSCrossRefMATHGoogle Scholar
  3. 3.
    Beltrán, J., Landim, C.: A Martingale approach to metastability. Probab. Theory Relat. Fields 161(1—-2), 267–307 (2015)CrossRefMATHGoogle Scholar
  4. 4.
    Landim, C.: A topology for limits of Markov chains. Stoch. Process. Appl. 125, 1058–1088 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Kramers, H.A.: Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284–304 (1940)MathSciNetADSCrossRefMATHGoogle Scholar
  6. 6.
    Freidlin, M.I., Wentzell, A.D.: Random perturbations of dynamical systems, 2nd edn. Translated from the 1979 Russian original by Joseph Szücs. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 260. Springer, New York (1998)Google Scholar
  7. 7.
    Galves, A., Olivieri, E., Vares, M.E.: Metastability for a class of dynamical systems subject to small random perturbations. Ann. Probab. 15, 1288–1305 (1987)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability in stochastic dynamics of disordered mean field models. Probab. Theory Relat. Fields 119, 99–161 (2001)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability and low-lying spectra in reversible Markov chains. Commun. Math. Phys. 228, 219–255 (2002)MathSciNetADSCrossRefMATHGoogle Scholar
  10. 10.
    Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times. J. Eur. Math. Soc. 6, 399–424 (2004)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bovier, A., Gayrard, V., Klein, M.: Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues. J. Eur. Math. Soc. 7, 69–99 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Cameron, M., Vanden-Eijnden, E.: Flows in complex networks: theory, algorithms, and application to LennardJones cluster rearrangement. J. Stat. Phys. 156, 427–454 (2014)MathSciNetADSCrossRefMATHGoogle Scholar
  13. 13.
    Noé, F., Wu, H., Prinz, J.H., Plattner, N.: Projected and hidden Markov models for calculating kinetics and metastable states of complex molecules (2013). arxiv:1309.3220v1
  14. 14.
    Cassandro, M., Galves, A., Olivieri, E., Vares, M.E.: Metastable behavior of stochastic dynamics: a pathwise approach. J. Stat. Phys. 35, 603–634 (1984)MathSciNetADSCrossRefMATHGoogle Scholar
  15. 15.
    Weinan, E., Vanden-Eijnden, E.: Towards a theory of transition paths. J. Stat. Phys. 123, 503–523 (2006)MathSciNetADSCrossRefMATHGoogle Scholar
  16. 16.
    Metzner, P., Schütte, Ch., Vanden-Eijnden, E.: Transition path theory for Markov jump processes. SIAM Multiscale Model. Simul. 7, 1192–1219 (2009)CrossRefMATHGoogle Scholar
  17. 17.
    Avena, L., Gaudillière, A.: On some random forests with determinantal roots (2013). arXiv:1310.1723v3
  18. 18.
    Bianchi, A., Bovier, A., Ioffe, D.: Sharp asymptotics for metastability in the random field Curie–Weiss model. Electron. J. Probab. 14, 1541–1603 (2009)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Bianchi, A., Bovier, A., Ioffe, D.: Pointwise estimates and exponential laws in metastable systems via coupling methods. Ann. Probab. 40, 339–371 (2012)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Gaudillière, A.: Condenser physics applied to Markov chains: a brief introduction to potential theory (2009). arXiv:0901.3053

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.IMPARio de JaneiroBrazil
  2. 2.CNRS UMR 6085, Université de RouenSaint-Étienne-du-RouvrayFrance
  3. 3.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

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