The percolation signature of the spin glass transition

J. MacHta; C. M. Newman; D. L. Stein

doi:10.1007/s10955-007-9446-2

The percolation signature of the spin glass transition

J. MacHta, C. M. Newman, D. L. Stein

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Magnetic ordering at low temperature for Ising ferromagnets manifests itself within the associated Fortuin-Kasteleyn (FK) random cluster representation as the occurrence of a single positive density percolating network. In this paper we investigate the percolation signature for Ising spin glass ordering-both in short-range (EA) and infinite-range (SK) models-within a two-replica FK representation and also within the different Chayes-Machta-Redner two-replica graphical representation. Based on numerical studies of the ±J EA model in three dimensions and on rigorous results for the SK model, we conclude that the spin glass transition corresponds to the appearance of two percolating clusters of unequal densities.

Original language	English (US)
Pages (from-to)	113-128
Number of pages	16
Journal	Journal of Statistical Physics
Volume	130
Issue number	1
DOIs	https://doi.org/10.1007/s10955-007-9446-2
State	Published - Jan 2008

Keywords

Cluster algorithms
Fortuin-Kasteleyn
Graphical representations
Ising spin glass
Percolation

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics

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10.1007/s10955-007-9446-2

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title = "The percolation signature of the spin glass transition",

abstract = "Magnetic ordering at low temperature for Ising ferromagnets manifests itself within the associated Fortuin-Kasteleyn (FK) random cluster representation as the occurrence of a single positive density percolating network. In this paper we investigate the percolation signature for Ising spin glass ordering-both in short-range (EA) and infinite-range (SK) models-within a two-replica FK representation and also within the different Chayes-Machta-Redner two-replica graphical representation. Based on numerical studies of the ±J EA model in three dimensions and on rigorous results for the SK model, we conclude that the spin glass transition corresponds to the appearance of two percolating clusters of unequal densities.",

keywords = "Cluster algorithms, Fortuin-Kasteleyn, Graphical representations, Ising spin glass, Percolation",

author = "J. MacHta and Newman, {C. M.} and Stein, {D. L.}",

note = "Funding Information: Acknowledgements We thank the Aspen Center for Physics where some of this work was performed. J.M. was supported in part by NSF DMR-0242402. He thanks New York University and Alan Sokal for hosting his sabbatical during which time this work was begun. C.M.N. was supported in part by NSF DMS-0102587 and DMS-0604869. D.L.S. was supported in part by NSF DMS-0102541 and DMS-0604869. Simulations were performed on the Courant Institute of Mathematical Sciences computer cluster.",

year = "2008",

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doi = "10.1007/s10955-007-9446-2",

language = "English (US)",

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AU - MacHta, J.

AU - Newman, C. M.

AU - Stein, D. L.

N1 - Funding Information: Acknowledgements We thank the Aspen Center for Physics where some of this work was performed. J.M. was supported in part by NSF DMR-0242402. He thanks New York University and Alan Sokal for hosting his sabbatical during which time this work was begun. C.M.N. was supported in part by NSF DMS-0102587 and DMS-0604869. D.L.S. was supported in part by NSF DMS-0102541 and DMS-0604869. Simulations were performed on the Courant Institute of Mathematical Sciences computer cluster.

PY - 2008/1

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N2 - Magnetic ordering at low temperature for Ising ferromagnets manifests itself within the associated Fortuin-Kasteleyn (FK) random cluster representation as the occurrence of a single positive density percolating network. In this paper we investigate the percolation signature for Ising spin glass ordering-both in short-range (EA) and infinite-range (SK) models-within a two-replica FK representation and also within the different Chayes-Machta-Redner two-replica graphical representation. Based on numerical studies of the ±J EA model in three dimensions and on rigorous results for the SK model, we conclude that the spin glass transition corresponds to the appearance of two percolating clusters of unequal densities.

AB - Magnetic ordering at low temperature for Ising ferromagnets manifests itself within the associated Fortuin-Kasteleyn (FK) random cluster representation as the occurrence of a single positive density percolating network. In this paper we investigate the percolation signature for Ising spin glass ordering-both in short-range (EA) and infinite-range (SK) models-within a two-replica FK representation and also within the different Chayes-Machta-Redner two-replica graphical representation. Based on numerical studies of the ±J EA model in three dimensions and on rigorous results for the SK model, we conclude that the spin glass transition corresponds to the appearance of two percolating clusters of unequal densities.

KW - Cluster algorithms

KW - Fortuin-Kasteleyn

KW - Graphical representations

KW - Ising spin glass

KW - Percolation

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The percolation signature of the spin glass transition

Abstract

Keywords

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