Spherical Sherrington–Kirkpatrick Model for Deformed Wigner Matrix with Fast Decaying Edges

Abstract

We consider the 2-spin spherical Sherrington–Kirkpatrick model whose disorder is given by a deformed Wigner matrix of the form \(W+\lambda V\), where W is a Wigner matrix and V is a random diagonal matrix with i.i.d. entries. Assuming that the density function of the entries of V decays faster than a certain rate near the edges of its spectrum, we prove the sharp phase transition of the limiting free energy and its fluctuation. In the high temperature regime, the fluctuation of the free energy converges in distribution to a Gaussian distribution, whereas it converges to a Weibull distribution in the low temperature regime. We also prove several results for deformed Wigner matrices, including a local law for the resolvent entries, a central limit theorem of the linear spectral statistics, and a theorem on the rigidity of eigenvalues.