{
  "annotation_id": "16c3da42-5283-4dad-a592-fc299455ad5f",
  "date_created": "2026-02-17T05:53:15.901000Z",
  "date_modified": "2026-02-17T05:53:15.901000Z",
  "file_hash": "2d7794fb92f657ea4de0324339ca01dbe0ef31411cde6bf89e7b21f9d9e0c97d",
  "private": false,
  "record": {
    "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",
      "math-ph",
      "math.MP",
      "stat.ML"
    ],
    "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"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "2eb61e43-1deb-4315-b724-a762c0709d38",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}