Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Stochastic Ising model with flipping sets of spins and fast decreasing temperature

Roy Cerqueti and Emilio De Santis

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This paper deals with the stochastic Ising model with a temperature shrinking to zero as time goes to infinity. A generalization of the Glauber dynamics is considered, on the basis of the existence of simultaneous flips of some spins. Such dynamics act on a wide class of graphs which are periodic and embedded in $\mathbb{R}^{d}$. The interactions between couples of spins are assumed to be quenched i.i.d. random variables following a Bernoulli distribution with support $\{-1,+1\}$. The specific problem here analyzed concerns the assessment of how often (finitely or infinitely many times, almost surely) a given spin flips. Adopting the classification proposed in (Comm. Math. Phys. 214 (2002) 373–387), we present conditions in order to have models of type $\mathcal{F}$ (any spin flips finitely many times), $\mathcal{I}$ (any spin flips infinitely many times) and $\mathcal{M}$ (a mixed case). Several examples are provided in all dimensions and for different cases of graphs. The most part of the obtained results holds true for the case of zero-temperature and some of them for the cubic lattice $\mathbb{L}_{d}=(\mathbb{Z}^{d},\mathbb{E}_{d})$ as well.


Cet article est dedié au modèle d’Ising stochastique avec une température decroissante à zéro avec le temps. Une généralisation de la dynamique de Glauber est considérée, basée sur des inversions simultanées des ensembles de spins. La dynamique est considerée sur une large classe de graphes qui sont périodiques et plongés dans un espace euclidien. Les interactions entre les couples de spins sont supposées être des variables aléatoires i.i.d. qui suivent une loi de Bernoulli avec support $\{-1,+1\}$. Le problème particulier analysé ici concerne l’évaluation du nombre d’inversions d’un spin donné (fini ou infini, presque sûrement). En adoptant la classification proposée dans (Comm. Math. Phys. 214 (2002) 373–387), nous présentons des conditions pour des modèles de type $\mathcal{F}$ (tout les spins sont sujets à un nombre fini d’inversions), $\mathcal{I}$ (tout les spins sont sujets à un nombre infini d’inversions) et $\mathcal{M}$ (le cas mixte). Plusieurs exemples sont fournis en toutes dimensions et pour plusieurs graphes. La partie majeure des résultats reste vraie à température zéro et certains des résultats sont vrais pour le réseau cubique en dimension $d$ quelconque.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 2 (2018), 757-789.

Received: 29 September 2015
Revised: 12 September 2016
Accepted: 16 January 2017
First available in Project Euclid: 25 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Ising model Glauber dynamics Fast decreasing temperature Graphs


Cerqueti, Roy; De Santis, Emilio. Stochastic Ising model with flipping sets of spins and fast decreasing temperature. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 2, 757--789. doi:10.1214/17-AIHP820.

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