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Zero-temperature Stochastic 2D Ising model and anisotropic curve-shortening flow

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The stochastic Ising model at zero temperature is the random evolution of a function {1,...,L}d \to {-,+}, the value of the function at one site is called spin. We start with the function that is uniformly equal to -1 on {1,...,L}d and let spin configuration evolve according to the following rule: At each step choose a site at random and change it spin to give it either the spin of the majority of its neighbors or +/- with probability p / (1-p) if there are equally many + and – in its neighborhood. When doing so we consider that sites that are outside of the cube are counted among the neighbors if needed and that they have a fixed + spin. With this procedure, the cube will eventually be filled with + spins. How many step do you need to perform in average so that this happens? In our talk we will give an answer for this question in both symmetric case (p=1/2) and asymmetric case (p>1/2) and we are able to describe precisely the scaling limit of the set of $-$ spins when time and space are rescaled. For the symmetric case we relate this result to a conjecture called “Lifshitz Law”. The stochastic Ising model (or Glauber Dynamics for Ising model), is a rather simplified model introduced by theoretical Physicists to study dynamical properties of ferromagnet.

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