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 $p\ge 3$ and $0<\beta \le {\beta }_{\mathrm{sh}}(p)$, we rule out shattering whenever $q\le 2^{-1/2}$ 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 Hölder's inequality give an additional $q$-dependent obstruction; in particular, they rule out every fixed overlap for $0<\beta \le \sqrt{\log 2}$. For $p=3$, the first two ranges already exhaust every fixed $q\in (0,1)$, so the landscape is not shattered at any $T\ge T_{\mathrm{sh}}$. For $p\ge 4$, the cases not covered by our criteria are confined to $2^{-1/2}<q\le \sqrt{(p-2)/(p-1)}$ and $\sqrt{\log 2}<\beta \le {\beta }_{\mathrm{sh}}(p)$. In particular, this paper partially resolves Conjecture 1 of the paper above and also suggests new methods to show non-shattering.

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