Spectral properties of a Non-Hermitian extension of the diluted Wishart ensemble

Abstract

We develop a theoretical framework based on the cavity and replica methods to analyze the spectral properties of sparse asymmetric correlation matrices of the form $\mathbf{F}=(\mathbf{X}{\mathbf{Y}}^{\top }+\omega \mathbf{Y}{\mathbf{X}}^{\top })/2T$, where $\mathbf{X}$ and $\mathbf{Y}$ are adjacency matrices of weighted Erdős--Rényi random graphs. We examine how the spectral density evolves as the asymmetry parameter $\omega$ varies from $0<\omega <1$ (nearly symmetric matrices) to $-1<\omega \le 0$ (nearly antisymmetric matrices). Analytical predictions are validated through exact numerical diagonalization, showing excellent agreement with theoretical results in the thermodynamic limit.

Updated author information; 22 pages, 2 figures