Convergence of the free energy for spherical spin glasses

We prove that the free energy of any spherical mixed \(p\)-spin model converges as the dimension \(N\) tends to infinity. While the convergence is a consequence of the Parisi formula, the proof we give is independent of the formula and uses the well-known Guerra-Toninelli interpolation method. The l...

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Bibliographic Details
Published inarXiv.org
Main Author Subag, Eliran
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 17.03.2022
Subjects
Approximation
Convergence
Free energy
Interpolation
Ising model
Mathematics - Mathematical Physics
Mathematics - Probability
Physics - Mathematical Physics
Spin glasses
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Summary:We prove that the free energy of any spherical mixed \(p\)-spin model converges as the dimension \(N\) tends to infinity. While the convergence is a consequence of the Parisi formula, the proof we give is independent of the formula and uses the well-known Guerra-Toninelli interpolation method. The latter was invented for models with Ising spins to prove that the free energy is super-additive and therefore (normalized by \(N\)) converges. In the spherical case, however, the configuration space is not a product space and the interpolation cannot be applied directly. We first relate the free energy on the sphere of dimension \(N+M\) to a free energy defined on the product of spheres in dimensions \(N\) and \(M\) to which we then apply the interpolation method. This yields an approximate super-additivity which is sufficient to prove the convergence.
ISSN:2331-8422
DOI:10.48550/arxiv.2203.09291