Free energy fluctuation of soft spherical Sherrington-Kirkpatrick model

We consider a soft version of the spherical Sherrington-Kirkpatrick model, where the spherical constraint is replaced by a radial confinement term. We prove that, for a real symmetric disorder matrix under suitable spectral assumptions, the free energy exhibits two distinct fluctuation regimes. In the high-temperature regime, the fluctuations are asymptotically Gaussian. In the low-temperature regime, the fluctuations are governed by the largest eigenvalue and converge to the GOE Tracy-Widom distribution. These results show that the soft radial constraint preserves the fluctuation transition of the classical spherical Sherrington-Kirkpatrick model.