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.

翻译：我们研究了具有独立同分布无序的纯与混合球形$p$-旋模型的自由能。在混合情形中，假设每个$p$-相互作用层要么具有指数为$α_p$的正则变化尾部，要么满足有限的$2p$阶矩条件。对于尾部指数为$α$的正则变化无序的纯球形$p$-旋模型，我们引入了一种尾部自适应的归一化方法，该方法在经典高斯尺度与极值尺度之间插值，并证明了淬火自由能的尖锐普适性二分定理。在亚临界区域$α<2p$中，热力学由有限多个极值耦合驱动，自由能收敛于一个由NIM（非相交单项式）模型描述的非退化随机极限，该极限仅依赖于极值次序统计量。在临界指数$α=2p$处，我们得到了一个随机的一维TAP型变分公式，该公式捕捉了球形切片上极值尖峰与普适高斯主体共存的现象。在超临界区域$α>2p$（更一般地，在有限$2p$阶矩假设下），自由能具有普适性，并与相应高斯模型的确定性Crisanti--Sommers/Parisi值一致，如[Sawhney-Sellke'24]中所建立。随后，我们将亚临界与临界结果推广至混合球形模型，其中每个$p$-层要么是满足$α_p\le 2p$的重尾分布，要么具有有限的$2p$阶矩。特别地，我们推导了混合模型的TAP型变分表示，从而实现了跨尾部指数与混合情形的淬火自由能的统一普适性分类。