The complexity of spherical $p$-spin models—A second moment approach

Abstract

Recently, Auffinger, Ben Arous and Černý 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 $\operatorname{Crt}_{N}(u)$, 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:

\[\mathbb{E}\{(\operatorname{Crt}_{N}(u))^{2}\}/(\mathbb{E} \{\operatorname{Crt}_{N}(u)\})^{2}\to1.\] As an immediate consequence we obtain that $\operatorname{Crt}_{N}(u)/\mathbb{E}\{\operatorname{Crt}_{N}(u)\}\to1$, in $L^{2}$, and thus in probability. For any $u$ for which $\mathbb{E}\operatorname{Crt}_{N}(u)$ does not tend to $0$ we prove that the moments match on an exponential scale.