Spherical Sherrington-Kirkpatrick model for deformed Wigner matrix with fast decaying edges

Understand

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 $F_N$ converges in distribution to a Gaussian distribution, whereas it converges to a Weibull distribution in the low temperature regime.