Fluctuations of the free energy of the mixed $p$-spin mean field spin glass model

Abstract

We prove the convergence in distribution of the fluctuations of the free energy of the mixed $p$-spin Sherrington-Kirkpatrick model with non-vanishing $2$-spin component at high enough temperature. The limit is Gaussian, and the fluctuations are seen to arise from weighted cycle counts in the complete graph on the spin indices weighted by the $2$-spin interaction matrix.