The Annals of Probability

Nucleation and growth for the Ising model in $d$ dimensions at very low temperatures

Raphaël Cerf and Francesco Manzo

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Abstract

This work extends to dimension $d\geq3$ the main result of Dehghanpour and Schonmann. We consider the stochastic Ising model on $\mathbb{Z}^{d}$ evolving with the Metropolis dynamics under a fixed small positive magnetic field $h$ starting from the minus phase. When the inverse temperature $\beta$ goes to $\infty$, the relaxation time of the system, defined as the time when the plus phase has invaded the origin, behaves like $\exp(\beta\kappa_{d})$. The value $\kappa_{d}$ is equal to

\[\kappa_{d}=\frac{1}{d+1}(\Gamma_{1}+\cdots+\Gamma_{d}),\]

where $\Gamma_{i}$ is the energy of the $i$-dimensional critical droplet of the Ising model at zero temperature and magnetic field $h$.

Article information

Source
Ann. Probab., Volume 41, Number 6 (2013), 3697-3785.

Dates
First available in Project Euclid: 20 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1384957776

Digital Object Identifier
doi:10.1214/12-AOP801

Mathematical Reviews number (MathSciNet)
MR3161463

Zentralblatt MATH identifier
1286.60091

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs

Keywords
Ising Metropolis metastability nucleation growth

Citation

Cerf, Raphaël; Manzo, Francesco. Nucleation and growth for the Ising model in $d$ dimensions at very low temperatures. Ann. Probab. 41 (2013), no. 6, 3697--3785. doi:10.1214/12-AOP801. https://projecteuclid.org/euclid.aop/1384957776


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