Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Statistical physics on a product of trees

Tom Hutchcroft

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Let $G$ be the product of finitely many trees $T_{1}\times T_{2}\times\cdots\times T_{N}$, each of which is regular with degree at least three. We consider Bernoulli bond percolation and the Ising model on this graph, giving a short proof that the model undergoes a second order phase transition with mean-field critical exponents in each case. The result concerning percolation recovers a result of Kozma (2013), while the result concerning the Ising model is new.

We also present a new proof, using similar techniques, of a lemma of Schramm concerning the decay of the critical two-point function along a random walk, as well as some generalizations of this lemma.


Soit $G$ le produit d’un nombre fini d’arbres $T_{1}\times T_{2}\times\cdots\times T_{N}$ ayant chacun un degré constant supérieur à trois, nous étudions la percolation de Bernoulli et le modèle d’Ising sur ce graphe et présentons une preuve simple de l’existence d’une transition de phase de second ordre ayant les mêmes exposants critiques que le modèle en champs moyen. Le résultat pour la percolation est une preuve alternative d’un résultat de Kozma (2013), tandis que le résultat pour le modèle d’Ising est nouveau.

Nous présentons également une nouvelle preuve, reposant sur des techniques similaires, du lemme de Schramm concernant la vitesse de décroissance de la fonction à deux points le long de la marche aléatoire, ainsi que des généralisations de ce lemme.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 1001-1010.

Received: 13 December 2017
Revised: 17 March 2018
Accepted: 13 April 2018
First available in Project Euclid: 14 May 2019

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Digital Object Identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60B99: None of the above, but in this section

Percolation Ising model Triangle condition Bubble diagram Nonuniqueness Nonamenable groups Mean-field Nonunimodular


Hutchcroft, Tom. Statistical physics on a product of trees. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 1001--1010. doi:10.1214/18-AIHP906.

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