Two conjectures about spectral density of diluted sparse Bernoulli random matrices

Abstract

We consider the ensemble of $N\times N$ ($N\gg 1$) symmetric random matrices with the bimodal independent distribution of matrix elements: each element could be either "1" with the probability $p$, or "0" otherwise. We pay attention to the "diluted" sparse regime, taking $p=1/N +\epsilon$, where $0<\epsilon \ll 1/N$. In this limit the eigenvalue density, $\rho(\lambda)$, is essentially singular, consisting of a hierarchical ultrametric set of peaks. We provide two conjectures concerning the structure of $\rho(\lambda)$: (i) we propose an equation for the position of sequential (in heights) peaks, and (ii) we give an expression for the shape of an outbound enveloping curve. We point out some similarities of $\rho(\lambda)$ with the shapes constructed on the basis of the Dedekind modular $\eta$-function.