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

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 Crt_N(u), they computed the asymptotics of 1/N log(E[double-struck]Crt_N(u)), as N, the dimension of the sphere, goes to∞.We compute the asymptotics of the corresponding second moment and show that, for p ≥ 3 and sufficiently negative u, it matches the first moment: E[double-struck] [(Crt_N(u))²] / (E[double-struck] [Crt_N(u)])²→1. As an immediate consequence we obtain that Crt_N(u)/E[double-struck][Crt_N(u)]→1, in L², and thus in probability. For any u for which E[double-struck]Crt_N(u) does not tend to 0 we prove that the moments match on an exponential scale.