Two conjectures about spectral density of diluted sparse Bernoulli random matrices

TL;DR

This paper investigates the spectral density of large sparse Bernoulli random matrices in a diluted regime, proposing conjectures about the structure and shape of the eigenvalue distribution, revealing hierarchical peaks and connections to modular functions.

Contribution

It introduces two conjectures describing the peak positions and envelope shape of the spectral density in a diluted sparse regime of Bernoulli matrices, advancing understanding of their spectral structure.

Findings

Spectral density exhibits hierarchical ultrametric peaks.

Proposed equations for peak positions and envelope shape.

Connections to Dedekind eta-function shape patterns.

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.