An exposition to information percolation for the Ising model

Lubetzky, Eyal; Sly, Allan

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 4, pp. 745-761.

La percolation de l’information est une nouvelle méthode pour analyser les systèmes de spins stochastiques à travers la classification et le contrôle des amas de flots d’information dans des tranches d’espace-temps. Elle fournit des estimées fines de mélange (transition abrupte dans une fenêtre d’ordre $O\left(1\right)$) pour le modèle d’Ising sur ${\mathbb{Z}}^d$ jusqu’à la température critique, ainsi que des résultats sur l’influence des conditions initiales sur le mélange. Dans cet article de présentation, nous appliquons cette méthode à des réseaux (plus généralement, sur tout graphe localement fini et transitif) à très haute température.

Information percolation is a new method for analyzing stochastic spin systems through classifying and controlling the clusters of information-flow in the space-time slab. It yielded sharp mixing estimates (cutoff with an $O\left(1\right)$-window) for the Ising model on ${\mathbb{Z}}^d$ up to the critical temperature, as well as results on the effect of initial conditions on mixing. In this expository note we demonstrate the method on lattices (more generally, on any locally-finite transitive graph) at very high temperatures.

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