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

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

2011-02-08

arxiv.org/abs/1102.1741v1

Mathematics

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Scientific paper

This work extends to dimensions $d\geq 3$ the main result of Dehghanpour and Schonmann. We consider the stochastic Ising model on $\Z^d$ 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 $$\kappa_d\,=\,\frac{1}{d+1}\big(\Gamma_1+...+\Gamma_d\big)$$ where $\Gamma_i$ is the energy of the $i$ dimensional critical droplet of the Ising model at zero temperature and magnetic field~$h$.

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