dorsal/arxiv
View SchemaA Sharp Universality Dichotomy for the Free Energy of Spherical Spin Glasses
| Authors | Taegyun Kim |
|---|---|
| Categories | |
| ArXiv ID | 2601.08599vv1 |
| URL | https://arxiv.org/abs/2601.08599 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
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.
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"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.\n 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 $\\alpha\u003c2p$, 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 $\\alpha\u003e2p$ (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\u002724].\n 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.",
"arxiv_id": "2601.08599",
"authors": [
"Taegyun Kim"
],
"categories": [
"math.PR",
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"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "A Sharp Universality Dichotomy for the Free Energy of Spherical Spin Glasses",
"url": "https://arxiv.org/abs/2601.08599",
"version": "v1"
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