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

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