Balanced multi-species spin glasses

TL;DR

This paper introduces a class of multi-species spin glass models with balanced interactions, providing a free energy lower bound applicable to various models, and demonstrates cases where this bound is tight, extending previous inequalities.

Contribution

It establishes a free energy lower bound for balanced multi-species spin glasses without convexity assumptions, generalizing prior results and identifying conditions for equality.

Findings

Lower bound matches the free energy of a single-species model.

Equality holds at high temperatures, for convex models, and at zero temperature in bipartite cases.

Connects spin glass free energy bounds to Gaussian tensor norms.

Abstract

We identify a special class of multi-species spin glass models: ones in which the species proportions serve to ''balance'' out the interaction strengths. For this class, we prove a free energy lower bound that does not require any convexity assumption, and applies to both Ising and spherical models. The lower bound is the free energy of a single-species model whose $p$-spin inverse-temperature parameter is exactly the variance of the $p$-spin component of the multi-species Hamiltonian. For the Ising case, this generalizes an inequality recently found by Issa in the context of vector spin models. We further demonstrate that this lower bound is actually an equality in many cases, including: at high temperatures for all models, at all temperatures for convex models, and at zero temperature for pure bipartite spherical models. When translated to a statement about the injective norm of a nonsymmetric Gaussian tensor, our lower bound matches an upper bound recently established by Dartois and McKenna.