Overlap distribution of spherical spin glass models with general eigenvalue distribution of the interaction matrix
Debapratim Banerjee, Debabrata Jana
TL;DR
The paper investigates how the edge-eigenvalue structure of a spiked interaction matrix drives replica symmetry in a spherical $2$-spin spin glass. By diagonalizing the GOE-based interaction, discretizing the semicircular bulk, and leveraging Dirichlet-coordinate representations together with a Laplace-approximation framework, it derives the exact limiting free energy per site for $\beta J>1$ and $J>1$, and characterizes the asymptotic overlap distribution between two Gibbs samples. The overlap limit reveals a continuous density at low temperature, signaling full replica symmetry breaking, with a concrete mixture of Gaussian-type components governed by two random variables $s_1,s_2$ and Rademacher factors. This behavior contrasts with one-outlier or ferromagnetic SK-CW scenarios and suggests limitations of the Parisi approach for this class of spherical spin glasses.
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.