Critical Fluctuations for the Spherical Sherrington-Kirkpatrick Model in an External Field

We prove the existence of a critical regime for the fluctuations of the ground-state energy of the spherical Sherrington-Kirkpatrick model in an external field, confirming predictions given in [3,12]. We also establish a critical regime for the fluctuations in a model with a critical Ferromagnetic interaction term, producing a three-parameter family of distributions generalizing the two-parameter family given in [4]. These results are both established in the generality of a $\beta$-ensemble analogue of the spherical Sherrington-Kirkpatrick model, which subsumes the complex and quarternionic generalizations.