The convex structure of the Parisi formula for multi-species spin glasses
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911098819575808 |
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| author | Chen, Hong-Bin Issa, Victor Mourrat, Jean-Christophe |
| author_facet | Chen, Hong-Bin Issa, Victor Mourrat, Jean-Christophe |
| contents | We study the free energy of mean-field multi-species spin glasses with convex covariance function. For such models with $D$ species, the Parisi formula is known to be valid, and expresses the limit free energy as a supremum over monotone probability measures on $\mathbb{R}_+^D$. We show here that one can transform this representation into a supremum over all probability measures on $\mathbb{R}_+^D$ of a concave functional. We then deduce that the Parisi formula admits a unique maximizer. Using convex-duality arguments, we also obtain a new representation of the free energy as an infimum over martingales in a Wiener space. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_06397 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The convex structure of the Parisi formula for multi-species spin glasses Chen, Hong-Bin Issa, Victor Mourrat, Jean-Christophe Probability Disordered Systems and Neural Networks 82D30, 82B44, 35D40 We study the free energy of mean-field multi-species spin glasses with convex covariance function. For such models with $D$ species, the Parisi formula is known to be valid, and expresses the limit free energy as a supremum over monotone probability measures on $\mathbb{R}_+^D$. We show here that one can transform this representation into a supremum over all probability measures on $\mathbb{R}_+^D$ of a concave functional. We then deduce that the Parisi formula admits a unique maximizer. Using convex-duality arguments, we also obtain a new representation of the free energy as an infimum over martingales in a Wiener space. |
| title | The convex structure of the Parisi formula for multi-species spin glasses |
| topic | Probability Disordered Systems and Neural Networks 82D30, 82B44, 35D40 |
| url | https://arxiv.org/abs/2508.06397 |