Generalizations of Parisi’s replica symmetry breaking and overlaps in random energy models
The random energy model (REM) is the simplest spin glass model which exhibits replica symmetry breaking. It is well known since the 80’s that its overlaps are non-selfaveraging and that their statistics satisfy the predictions of the replica theory. All these statistical properties can be understood...
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Académie des sciences
2024-09-01
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| Series: | Comptes Rendus. Physique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.199/ |
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| author | Derrida, Bernard Mottishaw, Peter |
| author_facet | Derrida, Bernard Mottishaw, Peter |
| author_sort | Derrida, Bernard |
| collection | DOAJ |
| description | The random energy model (REM) is the simplest spin glass model which exhibits replica symmetry breaking. It is well known since the 80’s that its overlaps are non-selfaveraging and that their statistics satisfy the predictions of the replica theory. All these statistical properties can be understood by considering that the low energy levels are the points generated by a Poisson process with an exponential density. Here we first show how, by replacing the exponential density by a sum of two exponentials, the overlaps statistics are modified. One way to reconcile these results with the replica theory is to allow the blocks in the Parisi matrix to fluctuate. Other examples where the sizes of these blocks should fluctuate include the finite size corrections of the REM, the case of discrete energies and the overlaps between two temperatures. In all these cases, the block sizes not only fluctuate but need to take complex values if one wishes to reproduce the results of our replica-free calculations. |
| format | Article |
| id | doaj-art-13b0dbdeb5f949b9afaa5e7e7dbcf9f2 |
| institution | Kabale University |
| issn | 1878-1535 |
| language | English |
| publishDate | 2024-09-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Physique |
| spelling | doaj-art-13b0dbdeb5f949b9afaa5e7e7dbcf9f22025-02-07T13:53:46ZengAcadémie des sciencesComptes Rendus. Physique1878-15352024-09-0125G132935110.5802/crphys.19910.5802/crphys.199Generalizations of Parisi’s replica symmetry breaking and overlaps in random energy modelsDerrida, Bernard0https://orcid.org/0000-0001-6994-0226Mottishaw, Peter1https://orcid.org/0000-0002-0091-4094Collège de France, 11 place Marcelin Berthelot, 75005 Paris, France; Laboratoire de Physique de l’Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, FranceSUPA, School of Physics and Astronomy, University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United KingdomThe random energy model (REM) is the simplest spin glass model which exhibits replica symmetry breaking. It is well known since the 80’s that its overlaps are non-selfaveraging and that their statistics satisfy the predictions of the replica theory. All these statistical properties can be understood by considering that the low energy levels are the points generated by a Poisson process with an exponential density. Here we first show how, by replacing the exponential density by a sum of two exponentials, the overlaps statistics are modified. One way to reconcile these results with the replica theory is to allow the blocks in the Parisi matrix to fluctuate. Other examples where the sizes of these blocks should fluctuate include the finite size corrections of the REM, the case of discrete energies and the overlaps between two temperatures. In all these cases, the block sizes not only fluctuate but need to take complex values if one wishes to reproduce the results of our replica-free calculations.https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.199/Disordered systemsSpin glassesReplica symmetry breakingRandom Energy Model |
| spellingShingle | Derrida, Bernard Mottishaw, Peter Generalizations of Parisi’s replica symmetry breaking and overlaps in random energy models Comptes Rendus. Physique Disordered systems Spin glasses Replica symmetry breaking Random Energy Model |
| title | Generalizations of Parisi’s replica symmetry breaking and overlaps in random energy models |
| title_full | Generalizations of Parisi’s replica symmetry breaking and overlaps in random energy models |
| title_fullStr | Generalizations of Parisi’s replica symmetry breaking and overlaps in random energy models |
| title_full_unstemmed | Generalizations of Parisi’s replica symmetry breaking and overlaps in random energy models |
| title_short | Generalizations of Parisi’s replica symmetry breaking and overlaps in random energy models |
| title_sort | generalizations of parisi s replica symmetry breaking and overlaps in random energy models |
| topic | Disordered systems Spin glasses Replica symmetry breaking Random Energy Model |
| url | https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.199/ |
| work_keys_str_mv | AT derridabernard generalizationsofparisisreplicasymmetrybreakingandoverlapsinrandomenergymodels AT mottishawpeter generalizationsofparisisreplicasymmetrybreakingandoverlapsinrandomenergymodels |