An exposition to information percolation for the Ising model

Lubetzky, Eyal; Sly, Allan

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Numéro Spécial : Conférence “Talking Across Fields” du 24 au 28 mars 2014 à l’Institut de Mathématiques de Toulouse, 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.

[1] Aldous (D.).— Random walks on finite groups and rapidly mixing Markov chains, Seminar on probability, XVII, p. 243-297 (1993). | Numdam | MR | Zbl

[2] Aldous (D.) and Diaconis (P.).— Shuffing cards and stopping times, Amer. Math. Monthly 93, p. 333-348 (1986). | MR | Zbl

[3] Diaconis (P.).— The cutoff phenomenon in finite Markov chains, Proc. Nat. Acad. Sci. U.S.A. 93, no. 4, p. 1659-1664 (1996). | MR | Zbl

[4] Diaconis (P.) and Shahshahani (M.).— Generating a random permutation with random transpositions, Z. Wahrsch. Verw. Gebiete 57, no. 2, p. 159-179 (1981). | MR | Zbl

[5] Liggett (T. M.).— Interacting particle systems, Classics in Mathematics, Springer-Verlag, Berlin (2005). | MR | Zbl

[6] Lubetzky (E.) and Sly (A.).— Cutoff phenomena for random walks on random regular graphs, Duke Math. J. 153, no. 3, p. 475-510 (2010). | MR | Zbl

[7] Lubetzky (E.) and Sly (A.).— Explicit expanders with cutoff phenomena, Electron. J. Probab. 16, no. 15, p. 419-435 (2011). | MR | Zbl

[8] Lubetzky (E.) and Sly (A.).— Cutoff for the Ising model on the lattice, Invent. Math. 191, no. 3, p. 719-755 (2013). | MR | Zbl

[9] Lubetzky (E.) and Sly (A.).— Cutoff for general spin systems with arbitrary boundary conditions, Comm. Pure. Appl. Math. 67, no. 6, p. 982-1027 (2014). | MR | Zbl

[10] Lubetzky (E.) and Sly (A.).— Information percolation and cutoff for thr stochastic Ising model. J. Amer. Math. Soc., to appear.

[11] Lubetzky (E.) and Sly (A.).— Universality of cutoff for the Ising model, preprint. Available at arXiv:1407.1761 (2014). | MR

[12] Martinelli (F.).— Lectures on Glauber dynamics for discrete spin models, Lectures on probability theory and statistics (Saint-Flour, 1997), Lecture Notes in Math., vol. 1717, Springer, Berlin, p. 93-191 (1999). | MR | Zbl

[13] Miller (J.) and Peres (Y.).— Uniformity of the uncovered set of random walk and cutoff for lamplighter chains, Ann. Probab. 40, no. 2, p. 535-577 (2012). | MR | Zbl

[14] Propp (J. G.) and Wilson (D. B.).— Exact sampling with coupled Markov chains and applications to statistical mechanics, Random Structures Algorithms 9, no. 1-2, p. 223-252 (1996). | MR | Zbl