Article Dans Une Revue Probability and Mathematical Physics Année : 2025

On the free energy of vector spin glasses with non-convex interactions

Résumé

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.

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Dates et versions

hal-04290374 , version 1 (16-11-2023)

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Hong-Bin Chen, Jean-Christophe Mourrat. On the free energy of vector spin glasses with non-convex interactions. Probability and Mathematical Physics, 2025, 6 (1), pp.1-80. ⟨10.2140/pmp.2025.6.1⟩. ⟨hal-04290374⟩
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