A new look at the interfaces in the percolation and Ising models

Abstract : The interfaces in the percolation and Ising models play an important role in the understanding of these models and are at the heart of several problematics: the Wulff construction, the mean curvature motion and the SLE theory. In his famous 1972 paper, Roland Dobrushin showed that the Ising model in dimensions d ≥ 3 has a Gibbs measure which is not invariant by translation by studying the interface between the top and the bottom of a straight finite box. The case of a tilted box is very different and more difficult to analyse. In this thesis, we propose a new definition of the interface. This definition is constructed in the Bernoulli percolation model with the help of a dynamical coupling between two configurations. We show that this interface is localized around the pivotal edges within a distance of order ln²n inside a box of size n. The proof relies on space-time paths which allow us to control the speed of the interface. We also show that the speed of the pivotal edges is at most of order ln n. We extend these results to the FK-percolation model, we also show the localization of the interface at distance of order ln²n around the pivotal edges. Using a modification of the classical Edwards-Sokal coupling, we obtain analogous results on the localization of the interface in the Ising model.
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Wei Zhou. A new look at the interfaces in the percolation and Ising models. Probability [math.PR]. Université Paris-Saclay, 2019. English. ⟨NNT : 2019SACLS173⟩. ⟨tel-02191676⟩

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