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

TL;DR

The paper develops a non-Hermitian extension of the diluted Wishart ensemble by studying sparse, asymmetric cross-correlation matrices $\boldsymbol{F}$ with $\boldsymbol{F} = (\boldsymbol{X}\boldsymbol{Y}^{\top} + \omega\boldsymbol{Y}\boldsymbol{X}^{\top})/(2d)$ and $\omega\in(-1,1)$. Using cavity and replica methods, it derives self-consistent equations for the spectral density in the thermodynamic limit and validates them against large-scale numerical diagonalization across regimes ranging from nearly antisymmetric to nearly Hermitian. The Edwards–Jones framework is adapted to the non-Hermitian, diluted setting, and both cavity and replica formalisms yield equivalent expressions for the average density $\langle \rho(z) \rangle_{\boldsymbol{F}}$. The results reveal how $\omega$, $q=N/T$, and $c=\mathbb{E}(x_{it}y_{it})$ shape the spectrum, including real- versus complex-valued spectra and boundedness near the origin in certain regimes, with practical advantages in accessing the thermodynamic-limit density without diagonalization.

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 $\boldsymbol{F} = (\boldsymbol{X}\boldsymbol{Y}^\top + ω\boldsymbol{Y}\boldsymbol{X}^\top)/2T$, where $\boldsymbol{X}$ and $\boldsymbol{Y}$ are adjacency matrices of weighted Erdős--Rényi random graphs. We examine how the spectral density evolves as the asymmetry parameter $ω$ varies from $0 < ω< 1$ (nearly symmetric matrices) to $-1 < ω\le 0$ (nearly antisymmetric matrices). Analytical predictions are validated through exact numerical diagonalization, showing excellent agreement with theoretical results in the thermodynamic limit.

Figures (2)

• Figure 1: Average spectral density of $\bm F$ from Eq. \ref{['eq:defF']} for different parameter choices. First row: $q=1$, $d=5$, $c=0.8$, $\omega=-0.8$; second row: $q=1$, $d=5$, $c=0.3$, $\omega=0.3$; third row: $q=1/2$, $d=5$, $c=0.1$, $\omega=-0.2$. Left panels: surfaces from numerical diagonalization (points) with theoretical predictions overlaid as solid coloured curves. Right panels: cuts of the spectral density along specific directions—first row: $\mathrm{Re}\,z=0.075$ (blue), $\mathrm{Re}\,z=0.125$ (green); second row: $\mathrm{Im}\,z=0.025$ (green), $\mathrm{Im}\,z=0.055$ (blue), $\mathrm{Im}\,z=0.085$ (purple); third row: $\mathrm{Re}\,z=0.05$ (blue), $\mathrm{Re}\,z=0.35$ (green), $\mathrm{Re}\,z=0.65$ (purple). In all cases, $300$ matrices are sampled, yielding $6\times10^6$ eigenvalues. Matrix sizes: $N=T=20\,000$ for the first and second rows; $N=T/2=20\,000$ for the third row.
• Figure 2: Finite-size effects in the spectral density for the parameters of the second row of Fig. \ref{['fig:omegaComparisons']}, along the cut $\mathrm{Im}\,z=0.085$. Red: numerical diagonalization with $40\,000\times40\,000$ matrices; green: $1000\times1000$ matrices; purple: theoretical prediction. The agreement improves with increasing $N$, consistent with convergence to the thermodynamic limit.