A Sharp Universality Dichotomy for the Free Energy of Spherical Spin Glasses

TL;DR

This paper establishes a sharp universality dichotomy for the free energy of spherical p-spin glasses, showing how tail behavior of disorder influences the thermodynamic limit and connecting extremal statistics with Gaussian universality.

Contribution

It introduces a tail-adapted normalization and a universality classification for the free energy of spherical p-spin models with heavy-tailed and finite-moment disorders.

Findings

In the subcritical regime, free energy converges to a non-degenerate limit driven by extremal couplings.

At the critical tail index, a TAP-type variational formula captures coexistence phenomena.

In the supercritical regime, free energy matches the Gaussian model's Parisi value, confirming universality.

Abstract

We study the free energy for pure and mixed spherical $p$-spin models with i.i.d.\ disorder. In the mixed case, each $p$-interaction layer is assumed either to have regularly varying tails with exponent $\alpha_p$ or to satisfy a finite $2p$-th moment condition. For the pure spherical $p$-spin model with regularly varying disorder of tail index $\alpha$, we introduce a tail-adapted normalization that interpolates between the classical Gaussian scaling and the extreme-value scale, and we prove a sharp universality dichotomy for the quenched free energy. In the subcritical regime $\alpha<2p$, the thermodynamics is driven by finitely many extremal couplings and the free energy converges to a non-degenerate random limit described by the NIM (non-intersecting monomial) model, depending only on extreme-order statistics. At the critical exponent $\alpha=2p$, we obtain a random one-dimensional TAP-type variational formula capturing the coexistence of an extremal spike and a universal Gaussian bulk on spherical slices. In the supercritical regime $\alpha>2p$ (more generally, under a finite $2p$-th moment assumption), the free energy is universal and agrees with the deterministic Crisanti--Sommers/Parisi value of the corresponding Gaussian model, as established in [Sawhney-Sellke'24]. We then extend the subcritical and critical results to mixed spherical models in which each $p$-layer is either heavy-tailed with $\alpha_p\le 2p$ or has finite $2p$-th moment. In particular, we derive a TAP-type variational representation for the mixed model, yielding a unified universality classification of the quenched free energy across tail exponents and mixtures.