Overlap distribution of spherical spin glass models with general eigenvalue distribution of the interaction matrix

Abstract

In this paper, we show that the replica symmetry of the Gibbs measure of spherical spin systems is a property of the eigenvalue spacing at the edge of the interaction matrix. In particular, our interaction matrix has \textbf{two} large outlier eigenvalues with mutual distance $\frac{c}{n}$. The empirical measure of the rest of the eigenvalues is close to the semicircular law with some rigidity conditions. We prove that in this scenario the overlap distribution of two independent samples from the Gibbs measure has a continuous density at a low enough temperature. Hence, the model is a full replica symmetry-breaking model. One might compare this result with only one outlier eigenvalue. This model comes for the Sherrington-Kirkpatrick model with Curie-Weiss interaction in the ferromagnetic case. Here, it is well known that the model is replica symmetric, although the free energy limit of this model is the same as the free energy limit of our model. In our limited understanding, we believe that this kind of phenomenon cannot be explained by the Parisi approach.