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

arXiv: https://arxiv.org/abs/2601.08599 · HTML: https://mlsanity.com/paper/2601.08599/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 36 references that are in the map.
- First posted on arXiv 13 Jan 2026.

## Builds on (most cited first, 5 of 5)

- [Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning](https://mlsanity.com/paper/1810.01075/) (2018)
- [A Tail-Index Analysis of Stochastic Gradient Noise in Deep Neural Networks](https://mlsanity.com/paper/1901.06053/) (2019)
- [A statistical model for tensor PCA](https://mlsanity.com/paper/1411.1076/) (2014)
- [Emergence of heavy tails in homogenized stochastic gradient descent](https://mlsanity.com/paper/2402.01382/) (2024)
- [Global law of conjugate kernel random matrices with heavy-tailed weights](https://mlsanity.com/paper/2502.18428/) (2025)

## Abstract (arXiv)

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 αp or to satisfy a finite 2p-th moment condition. For the pure spherical p-spin model with regularly varying disorder of tail index α, 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 α<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 α=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 α>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 αp≤ 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.