Free energy in multi-species mixed $p$-spin spherical models

Abstract

We prove a Parisi formula for the limiting free energy of multi-species spherical spin glasses with mixed $p$-spin interactions. The upper bound involves a Guerra-style interpolation and requires a convexity assumption on the model's covariance function. Meanwhile, the lower bound adapts the cavity method of Chen so that it can be combined with the synchronization technique of Panchenko; this part requires no convexity assumption. In order to guarantee that the resulting Parisi formula has a minimizer, we formalize the pairing of synchronization maps with overlap measures so that the constraint set is a compact metric space. This space is not related to the model's spherical structure and can be carried over to other multi-species settings.

75 pages; updated assumption (H2) for greater generality

References in corpus (7)

• Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
• On the overlap in the multiple spherical SK models
• Free energy and complexity of spherical bipartite models
• Annealing and replica-symmetry in Deep Boltzmann Machines
• Crisanti-Sommers formula and simultaneous symmetry breaking in multi-species spherical spin glasses
• Parisi's formula is a Hamilton-Jacobi equation in Wasserstein space
• Pattern recognition in Deep Boltzmann machines

Cited by in corpus (6)

• Replica symmetry breaking in dense neural networks
• Crisanti-Sommers formula and simultaneous symmetry breaking in multi-species spherical spin glasses
• Low-rank Matrix Estimation with Inhomogeneous Noise
• Hamilton-Jacobi equations from mean-field spin glasses
• A remark on the spherical bipartite spin glass
• Free Energy Subadditivity for Symmetric Random Hamiltonians