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We consider the $2$-spin spherical Sherrington--Kirkpatrick model whosedisorder 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 acertain rate near the edges of its spectrum, we prove the sharp phasetransition of the limiting free energy and its fluctuation. In the hightemperature regime, the fluctuation of $F_N$ converges in distribution to aGaussian distribution, whereas it converges to a Weibull distribution in thelow temperature regime. We also prove several results for deformed Wignermatrices, including a local law for the resolvent entries, a central limittheorem of the linear spectral statistics, and a theorem on the rigidity ofeigenvalues.

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