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

TL;DR

This paper establishes a Parisi formula for the limiting free energy of multi-species spherical spin glasses with mixed p-spin interactions, combining advanced interpolation, cavity, and synchronization techniques.

Contribution

It introduces a novel proof of the Parisi formula for multi-species models, utilizing a convexity assumption for the upper bound and a cavity method for the lower bound, with a new framework for the overlap measure space.

Findings

Proves the Parisi formula for multi-species spherical spin glasses.

Develops a new approach combining Guerra-style interpolation and cavity methods.

Ensures the Parisi functional has a minimizer through a novel overlap measure framework.

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.