Abstract
We consider a Glauber dynamics associated with the Ising model on a large two-dimensional box with minus boundary conditions and in the limit of a vanishing positive external magnetic field. The volume of this box increases quadratically in the inverse of the magnetic field. We show that at subcritical temperature and for a large class of starting measures, including measures that are supported by configurations with macroscopic plus-spin droplets, the system rapidly relaxes to some metastable equilibrium—with typical configurations made of microscopic plus-phase droplets in a sea of minus spins—before making a transition at an asymptotically exponential random time towards equilibrium—with typical configurations made of microscopic minus-phase droplets in a sea of plus spins inside a large contour that separates this plus phase from the boundary. We get this result by bounding from above the local relaxation times towards metastable and stable equilibria. This makes possible to give a pathwise description of such a transition, to control the asymptotic behaviour of the mixing time in terms of soft capacities and to give estimates of these capacities.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Working in dimension two, the word “area” could have been more appropriate. We will follow the usage by referring to volumes and surfaces rather than areas and perimeters.
As long as \(\phi (B_+)\) is positive, the restricted ensemble \(\mu _{\Lambda _h, -, h}(\cdot | {{\mathcal {R}}})\) will be concentrated on the same kind of configurations, but, because some dynamical quantities will also play a role, we will get stronger results by taking \(B_+\) close to \(B_c\) rather than only asking for the positivity of \(\phi (B_+)\).
The index \({{\mathcal {Y}}}\), rather than Y, in the notation \({{\mathcal {L}}}_{{\mathcal {Y}}}\) might seem unnatural since the generator depends on the whole process and not only on the configuration space, but we are foreseeing here a later more natural notation, in accordance with [2].
References
Bianchi, A., Gaudillière, A.: Metastable states, quasi-stationary distributions and soft measures. Stoch. Process. Appl. 126(6), 1622–1680 (2016)
Bianchi, A., Gaudillière, A., Milanesi, P.: On soft capacities, quasi-stationary distributions and the pathwise approach to metastability. arXiv:1807.11233, (2018)
Chayes, J., Chayes, L., Schonmann, R.H.: Exponential decay of connectivities in the two-dimensional Ising model. J. Stat. Phys. 49, 433–445 (1987)
Cassandro, M., Galves, A., Olivieri, E., Vares, M.E.: Metastable behaviour of stochastic dynamics: a pathwise approach. J. Stat. Phys. 35, 603–634 (1984)
Dobrushin, R.L., Kotecký, R., Shlosman, S.B.: Wulff construction: a global shape from local interactions. AMS Translations series, Providence, R.I. (1992)
Dehghanpour, P., Schonmann, R.: A Nucleation-and-growth model. Prob. Theory Relat. Fields 107, 123–135 (1997)
Flanders, H.: A proof of Minkowski’s inequality for convex curves. Am. Math. Mon. 75, 581–593 (1968)
Ioffe, D.: Large deviations for the 2D Ising model: a lower bound without cluster expansions. J. Stat. Phys. 74, 411–432 (1994)
Ioffe, D.: Exact large deviations up to \(T_c\) for the Ising model in two dimensions. Prob. Theory Relat. Fields 102, 313–330 (1995)
Martinelli, F.: On the two dimensional Ising model in the phase coexistence region. J. Stat. Phys. 76(5), 1179–1246 (1994)
Osserman, R.: The isoperimetric inequality. Bull. Am. Math. Soc. 84, 1182–1238 (1978)
Osserman, R.: Bonnesen-style isoperimetric inequalities. Am. Math. Mon. 86, 1–29 (1979)
Olivieri, E., Vares, M.E.: Large Deviations and Metastability. Cambridge University Press, Cambridge (2005)
Pfister, C.E.: Large deviations and phase separation in the two-dimension Ising model. Helv. Phys. Acta 64, 953–1054 (1991)
Sinclair, A.: Improved bounds for mixing rates of Markov chains and multicommodity flows. Comb. Probab. Comput. 1, 351–370 (1992)
Schonmann, R.H., Shlosman, S.B.: Wulff droplets and the metastable relaxation time of the kinetic Ising model. Commun. Math. Phys. 194(2), 389–462 (1998)
Acknowledgements
M. E. V. thanks Roberto Schonmann for many long conversations on metastability, in particular for the hospitality at UCLA back in 1997, when she was studying the paper [16], and they discussed the difficulties to achieve a result along the line of the current paper; she also thanks Augusto Q. Teixeira for discussions on the subject matter of the paper, and Vladas Sidoravicius (in memoriam) for inspiring general discussions on metastability. A. G. and P. M. thank Julien Sohier for the fruitful discussions they had in Leiden when studying [16], on which much of this work is based. This was possible thanks to the kind hospitality of Leiden university, which hosted them for two fall seasons through Frank den Hollander’s ERC Advanced Grant 267356-VARIS. A. G. and P. M. also thank the kind hospitality of the Universidade Federal do Rio de Janeiro through Faperj E26/102.338/2013. M. E. V. acknowledges partial support of CNPq (Grant 305075/2016-0) and Faperj E-26/203.948/2016.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Yvan Velenik.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Gaudillière, A., Milanesi, P. & Vares, M.E. Asymptotic Exponential Law for the Transition Time to Equilibrium of the Metastable Kinetic Ising Model with Vanishing Magnetic Field. J Stat Phys 179, 263–308 (2020). https://doi.org/10.1007/s10955-019-02463-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-019-02463-5
Keywords
- Metastability
- Glauber dynamics
- Exponential law
- Relaxation time
- Quasi-stationary measures
- Potential theory