Asymptotic Exponential Law for the Transition Time to Equilibrium of the Metastable Kinetic Ising Model with Vanishing Magnetic Field

• Published in: Journal of statistical physics Vol. 179; no. 2; pp. 263 - 308
• Main Authors: Gaudillière, A., Milanesi, P., Vares, M. E.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-04-2020; Springer; Springer Nature B.V; Springer Verlag
• Subjects: Analysis; Asymptotic properties; Boundary conditions; Configurations; Droplets; Equilibrium; Ising model; Magnetic fields; Mathematical and Computational Physics; Mathematics; Physical Chemistry; Physics; Physics and Astronomy; Probability; Quantum Physics; Statistical Physics and Dynamical Systems; Theoretical; Two dimensional models; Secondary: 60J27; 60J75; Quasi-stationary measures; Potential theory; Glauber dynamics; Relaxation time; Primary: 82C20; 60J45; Metastability; Exponential law; exponential law; quasi-stationary mea- sures; relaxation time; potential theory; quasi-stationary measures
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-019-02463-5

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.