Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On the number of ground states of the Edwards–Anderson spin glass model

Louis-Pierre Arguin and Michael Damron

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Ground states of the Edwards–Anderson (EA) spin glass model are studied on infinite graphs with finite degree. Ground states are spin configurations that locally minimize the EA Hamiltonian on each finite set of vertices. A problem with far-reaching consequences in mathematics and physics is to determine the number of ground states for the model on $\mathbb{Z}^{d}$ for any $d$. This problem can be seen as the spin glass version of determining the number of infinite geodesics in first-passage percolation or the number of ground states in the disordered ferromagnet. It was recently shown by Newman, Stein and the two authors that, on the half-plane $\mathbb{Z}\times\mathbb{N}$, there is a unique ground state (up to global flip) arising from the weak limit of finite-volume ground states for a particular choice of boundary conditions. In this paper, we study the entire set of ground states on the infinite graph, proving that the number of ground states on the half-plane must be two (related by a global flip) or infinity. This is the first result on the entire set of ground states in a non-trivial dimension. In the first part of the paper, we develop tools of interest to prove the analogous result on $\mathbb{Z}^{d}$.


Les états fondamentaux du modèle de verre de spins de Edwards–Anderson (EA) sont étudiés sur des graphes infinis de degré fini. Les états fondamentaux sont les configurations de spins qui minimisent de manière locale l’Hamiltonien pour chaque ensemble fini de sommets. Un problème avec des implications importantes en physique et en mathématique est de déterminer le nombre d’états fondamentaux pour le modèle sur $\mathbb{Z}^{d}$ pour un $d>1$ donné. Ce problème est la version équivalente pour les modèles de verre de spins du problème du nombre de géodésiques infinies en percolation de premier passage et du nombre d’états fondamentaux du modèle d’Ising ferromagnétique désordonné. Il a été montré récemment par Newman, Stein et les deux auteurs que sur le demi-plan $\mathbb{Z}\times\mathbb{N}$, il existe un unique état fondamental (modulo un flip global des spins) produit par la limite faible des états fondamentaux des volumes finis pour un choix spécifique des conditions frontières. Dans cet article, nous étudions l’ensemble de tous les états fondamentaux sur le graphe infini $\mathbb{Z}\times\mathbb{N}$. Nous montrons que le nombre d’états fondamentaux est deux (correspondant à un flip global des spins) ou infini. Ceci est le premier résultat sur l’ensemble de tous les états fondamentaux pour une dimension non-triviale. Dans la première partie, nous développons des outils qui sont pertinents à la résolution du problème analogue sur $\mathbb{Z}^{d}$.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 1 (2014), 28-62.

First available in Project Euclid: 1 January 2014

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Primary: 82D30: Random media, disordered materials (including liquid crystals and spin glasses) 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Spin glasses Edwards–Anderson model Ground states


Arguin, Louis-Pierre; Damron, Michael. On the number of ground states of the Edwards–Anderson spin glass model. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 1, 28--62. doi:10.1214/12-AIHP499.

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