The complexity of spherical p-spin models - a second moment approach

TL;DR

This paper analyzes the second moment of the number of critical points in spherical p-spin models, showing that for certain parameters, the second moment matches the first, indicating concentration and providing insights into the model's complexity.

Contribution

It computes the asymptotics of the second moment of critical points in spherical p-spin models, extending previous first moment results and establishing moment matching for certain parameters.

Findings

Second moment asymptotics match the first moment for p≥3 and negative u.

Number of critical points concentrates around its expectation in probability.

Moments match exponentially on the scale where the expectation does not vanish.

Abstract

Recently, Auffinger, Ben Arous, and \v{C}ern\'y initiated the study of critical points of the Hamiltonian in the spherical pure $p$-spin spin glass model, and established connections between those and several notions from the physics literature. Denoting the number of critical values less than $Nu$ by $\mbox{Crt}_{N}(u)$, they computed the asymptotics of $\frac{1}{N}\log(\mathbb{E}\mbox{Crt}_{N}(u))$, as $N$, the dimension of the sphere, goes to $\infty$. We compute the asymptotics of the corresponding second moment and show that, for $p\geq3$ and sufficiently negative $u$, it matches the first moment: \[ \mathbb{E}\left\{ \left(\mbox{Crt}_{N}\left(u\right)\right)^{2}\right\} /\left(\vphantom{\left(\mbox{Crt}_{N}\left(u\right)\right)^{2}}\mathbb{E}\left\{ \mbox{Crt}_{N}\left(u\right)\right\} \right)^{2}\to1. \] As an immediate consequence we obtain that $\mbox{Crt}_{N}(u)/\mathbb{E}\{ \mbox{Crt}_{N}(u)\} \to 1$, in $L^2$ and thus in probability. For any $u$ for which $\mathbb{E}\mbox{Crt}_{N}(u)$ does not tend to $0$ we prove that the moments match on an exponential scale.