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 $\boldsymbol{F} = (\boldsymbol{X}\boldsymbol{Y}^\top + \omega \boldsymbol{Y}\boldsymbol{X}^\top)/2T$, where $\boldsymbol{X}$ and $\boldsymbol{Y}$ are adjacency matrices of weighted Erd\H{o}s--R\'enyi 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.