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 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.