Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick model with sparse interaction

We consider the spherical Sherrington-Kirkpatrick model of spin glass with sparse interaction, where the interactions between most of the pairs of the spin variables are possibly zero. With suitable normalization, we prove that the limiting free energy does not depend on the sparsity whereas the fluctuation of the free energy does. We also prove that both in the high- and the low-temperature regimes the fluctuation of the free energy converges in distribution to Gaussian distributions of same order when the sparsity is on a certain level, but their variances are different.