Recently, Auffinger, Ben Arous, and \v{C}ern\'y initiated the study ofcritical points of the Hamiltonian in the spherical pure $p$-spin spin glassmodel, and established connections between those and several notions from thephysics 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 thesphere, goes to $\infty$. We compute the asymptotics of the correspondingsecond moment and show that, for $p\geq3$ and sufficiently negative $u$, itmatches 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 immediateconsequence 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 momentsmatch on an exponential scale.
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