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

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 Holder's inequality give an additional q-dependent obstruction; in particular, they rule out every fixed overlap for 0<etaleqsqrt{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_{sh}. For pgeq4, the cases not covered by our criteria are confined to 2^{-1/2}<qleqsqrt{(p-2)/(p-1)} and sqrt{log2}<etaleqeta_{sh}(p). In particular, this paper partially resolves Conjecture 1 of the paper above and also suggests new methods to show non-shattering.

Source: arXiv cs.LG | 2026-08-17