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

Louis-Pierre Arguin; Michael Damron

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 1, page 28-62
  • ISSN: 0246-0203

Abstract

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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 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 ℤ × 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 ℤ d .

How to cite

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Arguin, Louis-Pierre, and Damron, Michael. "On the number of ground states of the Edwards–Anderson spin glass model." Annales de l'I.H.P. Probabilités et statistiques 50.1 (2014): 28-62. <http://eudml.org/doc/272052>.

@article{Arguin2014,
abstract = {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 $\mathbb \{Z\}^\{d\}$.},
author = {Arguin, Louis-Pierre, Damron, Michael},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {spin glasses; Edwards-Anderson model; ground states; multiple ground states; infinite graphs; first-passage percolation; infinite geodesics; half-plane uniqueness; measures on sets of ground states},
language = {eng},
number = {1},
pages = {28-62},
publisher = {Gauthier-Villars},
title = {On the number of ground states of the Edwards–Anderson spin glass model},
url = {http://eudml.org/doc/272052},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Arguin, Louis-Pierre
AU - Damron, Michael
TI - On the number of ground states of the Edwards–Anderson spin glass model
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 1
SP - 28
EP - 62
AB - 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 $\mathbb {Z}^{d}$.
LA - eng
KW - spin glasses; Edwards-Anderson model; ground states; multiple ground states; infinite graphs; first-passage percolation; infinite geodesics; half-plane uniqueness; measures on sets of ground states
UR - http://eudml.org/doc/272052
ER -

References

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