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

Recently, Auffinger, Ben Arous and .ern. 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 CrtN(u), they computed the asymptotics of 1Nlog(ECrtN(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{(CrtN(u))2}/(E{CrtN(u)})2.1. As an immediate consequence we obtain that CrtN(u)/E{CrtN(u)}.1, in L2, and thus in probability. For any u for which ECrtN(u) does not tend to 0 we prove that the moments match on an exponential scale.