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

Taegyun Kim

TL;DR

The paper establishes a sharp universality dichotomy for the quenched free energy of spherical spin glasses under heavy-tailed disorder, identifying subcritical, critical, and supercritical regimes relative to the tail exponent $\alpha$ and interaction order $p$. It introduces a tail-adapted normalization that interpolates between Gaussian scaling and extreme-value scaling, and uses a non-intersecting monomial (NIM) reduction to capture extremal-dominant behavior, together with a TAP-type variational formula at criticality that balances spike and bulk contributions. For mixed spherical models, the authors derive a unified TAP-type variational representation that accounts for mixtures of heavy-tailed and finite-moment layers, yielding a complete universality classification of the quenched free energy across tail exponents. The work connects extremes, bulk Parisi universality, and slice-wise covariance shifts to provide a comprehensive framework for understanding when extremal couplings govern thermodynamics versus when Gaussian Parisi behavior dominates, with potential implications for high-dimensional optimization and inference in heavy-tailed landscapes.

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 $α_p$ 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 $α_p\le 2p$ 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.

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

TL;DR

The paper establishes a sharp universality dichotomy for the quenched free energy of spherical spin glasses under heavy-tailed disorder, identifying subcritical, critical, and supercritical regimes relative to the tail exponent and interaction order . It introduces a tail-adapted normalization that interpolates between Gaussian scaling and extreme-value scaling, and uses a non-intersecting monomial (NIM) reduction to capture extremal-dominant behavior, together with a TAP-type variational formula at criticality that balances spike and bulk contributions. For mixed spherical models, the authors derive a unified TAP-type variational representation that accounts for mixtures of heavy-tailed and finite-moment layers, yielding a complete universality classification of the quenched free energy across tail exponents. The work connects extremes, bulk Parisi universality, and slice-wise covariance shifts to provide a comprehensive framework for understanding when extremal couplings govern thermodynamics versus when Gaussian Parisi behavior dominates, with potential implications for high-dimensional optimization and inference in heavy-tailed landscapes.

Abstract

We study the free energy for pure and mixed spherical -spin models with i.i.d.\ disorder. In the mixed case, each -interaction layer is assumed either to have regularly varying tails with exponent or to satisfy a finite -th moment condition. For the pure spherical -spin model with regularly varying disorder of tail index , 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 , 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 , 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 (more generally, under a finite -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 -layer is either heavy-tailed with or has finite -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.
Paper Structure (18 sections, 12 theorems, 87 equations)

This paper contains 18 sections, 12 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Model, normalization, and main results
  3. Spherical Mixed $p$-spin Model
  4. Regular variation and the heavy-tail quantile scale
  5. Gaussian Parisi/Crisanti--Sommers free energy
  6. A single-monomial free energy functional
  7. Main theorems (pure and mixed)
  8. Background: Gaussian Parisi theory and universality
  9. Parisi/Crisanti--Sommers formula
  10. Universality for light tails
  11. NIM Model and the monomial functional
  12. Proof of the main theorems
  13. Spike/bulk split and bulk moment bounds
  14. Negligibility of the Bulk Contribution $H_N^B$
  15. Size of the support for spike $H_N^A$
  16. ...and 3 more sections

Key Result

Theorem 2.4

Fix $p\ge2$, $\gamma_p=1$, and $\gamma_r=0$ for $r\neq p$, and consider the normalized Hamiltonian $\bar{H}_{N,p}$ as Definition def:HNbar. Assume that $H^{(p)}$ is $\alpha$-regularly varying. Fix $\beta\in(0,\infty)$. (i) Subcritical heavy tails $\alpha<2p$. Then, with $\Lambda_{N,p}$ as in eq:bNp, where $\Lambda$ is a Fréchet random variable of index $\alpha$ (the limit of $\Lambda_{N,p}$). (ii)

Theorems & Definitions (29)

  • Definition 2.1: Regularly varying tails
  • Definition 2.2
  • Definition 2.3: Monomial free energy functional
  • Theorem 2.4: Pure $p$-spin: sharp universality dichotomy and critical competition
  • Remark 2.5
  • Theorem 2.6: Mixed model: extremal terms vs. Parisi bulk
  • Remark 2.7
  • Proposition 3.1: Adaptation of Gaussian universality
  • proof : Proof sketch
  • Lemma 3.2: Closed form for $f_p$
  • ...and 19 more