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

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.