Legendre Duality of Spherical and Gaussian Spin Glasses

Abstract

The classical result of concentration of the Gaussian measure on the sphere in the limit of large dimension induces a natural duality between Gaussian and spherical models of spin glass. We analyse the Legendre variational structure linking the free energies of these two systems, in the spirit of the equivalence of ensembles of statistical mechanics. Our analysis, combined with the previous work [4], shows that such models are replica symmetric. Lastly, we briefly discuss an application of our result to the study of the Gaussian Hopfield model.

References in corpus (6)

• Equilibrium statistical mechanics of bipartite spin systems
• On the overlap in the multiple spherical SK models
• Classical Particle in a Box with Random Potential: exploiting rotational symmetry of replicated Hamiltonian
• How glassy are neural networks?
• Universality in bipartite mean field spin glasses
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Cited by in corpus (10)

• Phase Diagram of Restricted Boltzmann Machines and Generalised Hopfield Networks with Arbitrary Priors
• Phase transitions in Restricted Boltzmann Machines with generic priors
• Non-Convex Multi-species Hopfield models
• High-dimensional rank-one nonsymmetric matrix decomposition: the spherical case
• The TAP-Plefka variational principle for the spherical SK model
• Legendre Equivalences of Spherical Boltzmann Machines
• Hopfield model with planted patterns: a teacher-student self-supervised learning model
• The effect of priors on Learning with Restricted Boltzmann Machines
• Whitney Smooth Families of Invariant Tori within the Reversible Context 2 of KAM Theory
• A remark on the spherical bipartite spin glass