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Preprints, Working Papers, ... Year : 2023

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

Hong-Bin Chen
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Abstract

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 and 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. 2023. ⟨hal-04290374⟩
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