Resumen de The complexity of spherical pp-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 pp-spin spin glass model, and established connections between those and several notions from the physics literature. Denoting the number of critical values less than NuNu by CrtN(u)CrtN(u), they computed the asymptotics of 1Nlog(ECrtN(u))1Nlog(ECrtN(u)), as NN, the dimension of the sphere, goes to ∞∞. We compute the asymptotics of the corresponding second moment and show that, for p≥3p≥3 and sufficiently negative uu, 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)}→1CrtN(u)/E{CrtN(u)}→1, in L2L2, and thus in probability. For any uu for which ECrtN(u)ECrtN(u) does not tend to 00 we prove that the moments match on an exponential scale.
