Non-Shattering at and Above the Dynamical Temperature in the Spherical Pure p-Spin Model

Abstract

We consider the notion of shattering introduced by Ben Arous and Jagannath for spherical pure $p$-spin glasses with overlap $q$. For every $p\geq 3$ and $0<\beta\leq\beta_{\mathrm{sh}}(p)$, we rule out shattering whenever $q\leq2^{-1/2}$ or $q>\sqrt{(p-2)/(p-1)}$. The proof combines a deterministic $N+1$ bound for disjoint bands in the first range with a general-$p$ sign law showing that their total marked weight has subdominant free energy in the second. A spherical-code bound and H\"older's inequality give an additional $q$-dependent obstruction; in particular, they rule out every fixed overlap for $0<\beta\leq\sqrt{\log2}$. For $p=3$, the first two ranges already exhaust every fixed $q\in(0,1)$, so the landscape is not shattered at any $T\geq T_{\mathrm{sh}}$. For $p\geq4$, the cases not covered by our criteria are confined to $2^{-1/2}<q\leq\sqrt{(p-2)/(p-1)}$ and $\sqrt{\log2}<\beta\leq\beta_{\mathrm{sh}}(p)$. In particular, this paper partially resolves Conjecture 1 of the paper above and also suggests new methods to show non-shattering.