aiqbench analysis
This research investigates the non-shattering of energy landscapes in spherical pure p-spin glasses, establishing rigorous bounds on overlap and temperature parameters where shattering is ruled out. By combining deterministic band analysis with general-p sign laws, the study partially resolves a prior conjecture regarding the stability of these complex systems above dynamical temperatures. These findings advance theoretical understanding of disordered systems relevant to spin glass theory and machine learning optimization landscapes.
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 $p\geq 3$ and $0\sqrt{(p-2)/(p-1)}$. 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.
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 Hölder'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.
25pages, Comments Welcome!
Probability (math.PR); Disordered Systems and Neural Networks (cond-mat.dis-nn); Machine Learning (cs.LG); Mathematical Physics (math-ph)
arXiv:2608.14369 [math.PR]
(or arXiv:2608.14369v1 [math.PR] for this version)
https://doi.org/10.48550/arXiv.2608.14369 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 251 words
