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

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 $\mathit{F}=(\mathit{X}{\mathit{Y}}^{\mathrm{\top }}+\omega \mathit{Y}{\mathit{X}}^{\mathrm{\top }})\mathrm{/}2T$, where $\mathit{X}$ and $\mathit{Y}$ 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.