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Critical Droplets and sharp asymptotics for Kawasaki dynamics with strongly anisotropic interactions

Mathematical Physics and Probability Seminar

Critical Droplets and sharp asymptotics for Kawasaki dynamics with strongly anisotropic interactions
Series: Mathematical Physics and Probability Seminar
Location: online
Presenter: Simone Baldassarri, University of Florence

In this talk we analyze metastability and nucleation in the context of the Kawasaki dynamics for the two-dimensional Ising lattice gas at very low temperature. Let $\Lambda\subset\mathbb{Z}^2$ be a finite box. Particles perform simple exclusion on $\Lambda$, but when they occupy neighboring sites they feel a binding energy $-U_1<0$ in the horizontal direction and $-U_2<0$ in the vertical one. Along each bond touching the boundary of $\Lambda$ from the outside to the inside, particles are created with rate $\rho=e^{-\Delta\beta}$, while along each bond from the inside to the outside, particles are annihilated with rate $1$, where $\beta>0$ is the inverse temperature and $\Delta>0$ is an activity parameter. We consider the parameter regime $U_1>2U_2$ also known as the strongly anisotropic regime. We take $\Delta\in{(U_1,U_1+U_2)}$, so that the empty (respectively full) configuration is a metastable (respectively stable) configuration. We consider the asymptotic regime corresponding to finite volume in the limit as $\beta\rightarrow\infty$.  We investigate how the transition from empty to full takes place with particular attention to the critical configurations that asymptotically have to be crossed with probability 1. To this end, we provide a model-independent strategy to identify some unessential saddles (that are not in the union of minimal gates) for the transition from the metastable (or stable) to the stable states and we apply this method to our model. The derivation of some geometrical properties of the saddles allows us to identify the full geometry of the minimal gates and their boundaries for the nucleation in the strongly anisotropic case. Moreover, we derive sharp estimates for the asymptotic transition time for the strongly anisotropic case. This is based on a joint work with F. R. Nardi.

(https://arizona.zoom.us/j/85470607369)