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

The Hamiltonian of the spherical p-spin spin glass model is a smooth Gaussian field on the N-dimensional sphere. Let $Crt_N(u)$ denote the number of its critical points below $Nu$. In a recent study Auffinger, Ben Arous, and Cerny computed the mean of $Crt_N(u)$ and its exponential growth rate, as N goes to infinity. Our work focuses on the computation of the second moment. We prove that the ratio of second to first moment squared goes to 1, as N goes to infinity. An immediate consequence of this is that $Crt_N(u)$ concentrates around its mean: $Crt_N(u)$ normalized by its mean goes to 1 in L^2 and thus in probability. Joint work with Ofer Zeitouni.