On the free energy of vector spin glasses with non-convex interactions
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866912818975997952 |
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| author | Chen, Hong-Bin Mourrat, Jean-Christophe |
| author_facet | Chen, Hong-Bin Mourrat, Jean-Christophe |
| contents | The limit free energy of spin-glass models with convex interactions can be represented as a variational problem involving an explicit functional. Models with non-convex interactions are much less well-understood, and simple variational formulas involving the same functional are known to be invalid in general. We show here that a slightly weaker property of the limit free energy does extend to non-convex models. Indeed, under the assumption that the limit free energy exists, we show that this limit can always be represented as a critical value of the said functional. Up to a small perturbation of the parameters defining the model, we also show that any subsequential limit of the law of the overlap matrix is a critical point of this functional. We believe that these results capture the fundamental conclusions of the non-rigorous replica method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_08980 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the free energy of vector spin glasses with non-convex interactions Chen, Hong-Bin Mourrat, Jean-Christophe Probability Disordered Systems and Neural Networks 82B44, 82D30 The limit free energy of spin-glass models with convex interactions can be represented as a variational problem involving an explicit functional. Models with non-convex interactions are much less well-understood, and simple variational formulas involving the same functional are known to be invalid in general. We show here that a slightly weaker property of the limit free energy does extend to non-convex models. Indeed, under the assumption that the limit free energy exists, we show that this limit can always be represented as a critical value of the said functional. Up to a small perturbation of the parameters defining the model, we also show that any subsequential limit of the law of the overlap matrix is a critical point of this functional. We believe that these results capture the fundamental conclusions of the non-rigorous replica method. |
| title | On the free energy of vector spin glasses with non-convex interactions |
| topic | Probability Disordered Systems and Neural Networks 82B44, 82D30 |
| url | https://arxiv.org/abs/2311.08980 |