arXiv:2608.14369v1 Announce Type: cross
Abstract: We consider the notion of shattering introduced by Ben Arous and Jagannath for spherical pure $p$-spin glasses with overlap $q$. For every $pgeq 3$ and $0sqrt{(p-2)/(p-1)}$. 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 $0<betaleqsqrt{log2}$. For $p=3$, the first two ranges already exhaust every fixed $qin(0,1)$, so the landscape is not shattered at any $Tgeq T_{mathrm{sh}}$. For $pgeq4$, the cases not covered by our criteria are confined to $2^{-1/2}<qleqsqrt{(p-2)/(p-1)}$ and $sqrt{log2}<betaleqbeta_{mathrm{sh}}(p)$. In particular, this paper partially resolves Conjecture 1 of the paper above and also suggests new methods to show non-shattering.