Fluctuations of the free energy of the Sherrington-Kirkpatrick model with ferromagnetic interaction

Abstract

We study the Sherrington-Kirkpatrick model with an additional Curie-Weiss ferromagnetic interaction (SKFI), whose phase diagram in the plane of the disorder strength $\beta$ and the ferromagnetic coupling $\gamma$ consists of a paramagnetic, a ferromagnetic, and a spin glass region. Our main result is a central limit theorem for the free energy in the ferromagnetic regime: at scale $N^{-1/2}$, the fluctuations coincide with those of the Sherrington-Kirkpatrick model in an effective external field determined by the limiting magnetization. We prove this for general mixed even $p$-spin interactions, using the fluctuation theory of Chen, Dey, and Panchenko. For the pure 2-spin model we complete the picture of the phase diagram: we give a new proof of the order-$N^{-1}$ Gaussian fluctuations in the paramagnetic regime, obtained earlier by Banerjee, valid up to the critical window; we determine the limiting distribution on the critical line $\gamma = 1 + \theta N^{-1/2}$ separating the paramagnetic and ferromagnetic regimes; and we show that in the spin glass regime the fluctuations coincide with those of the zero-field SK model under a widely believed variance-divergence assumption. Our results form the Ising analogue of the fluctuation results of Baik and Lee for the spherical SKFI model.