Mean eigenvector self-overlap in the real and complex elliptic Ginibre ensembles at strong and weak non-Hermiticity

Abstract

We study the mean diagonal overlap of left and right eigenvectors associated with complex eigenvalues in $N\times N$ non-Hermitian random Gaussian matrices. In well known works by Chalker and Mehlig the expectation of this (self-)overlap was computed for the complex Ginibre ensemble as $N\to \infty$. In the present work, we consider the same quantity in the real and complex elliptic Ginibre ensembles characterized by correlations between off-diagonal entries controlled by a parameter $τ\in[0,1]$, with $τ=1$ corresponding to the Hermitian limit. We derive exact expressions for the mean diagonal overlap in both ensembles at any finite $N$, for any eigenvalue off the real axis. We further investigate several scaling regimes as $N\rightarrow \infty$, both in the limit of strong non-Hermiticity keeping a fixed $τ\in[0,1)$ and in the weak non-Hermiticity limit, with $τ$ approaching unity in such a way that $N(1-τ)$ remains finite.

Figures (6)

• Figure 1: Large $N$ density of complex eigenvalues in the eGinOE (left) and eGinUE (right) in the strong (top) and (bottom) weak non-Hermiticity regimes with relevant scaling regions labelled. Strong non-Hermiticity plots are shown for $\tau = 0.25$ and weak non-Hermiticity plots are for $\alpha = 1$. In all plots the solid black ellipse is given by Eq. \ref{['eq:ellipse']} and the density is depicted on a rainbow scale. This diagram explains schematically what is meant by the bulk, edge and depletion regimes of large eGinOE and eGinUE matrices. Each heatmap is plotted using the finite $N$ expression for the density of complex eigenvalues in the associated ensembles when $N=125$.
• Figure 2: Normalised conditional density of complex eigenvalues, $\Tilde{\rho}_{\text{WNH}}$, see Eq.(\ref{['condden']}), at WNH in the eGinOE (green) and eGinUE (red). Left: $\Tilde{\rho}_{\text{WNH}}$ as a function of y, at fixed $X=0$ and $\alpha = 2$. Right: $\Tilde{\rho}_{\text{WNH}}$ as a function of X, at fixed $y=1$ and $\alpha = 1$. Each plot contains a normalised histogram of eigenvalues within $\pm 1/\sqrt{N}$ of the fixed $X$ or $y$, selected from an original set containing $O(10^8)$ eigenvalues.
• Figure 3: Mean conditional self-overlap, $\mathbb{E}_N(x,y)$, see Eq.(\ref{['conditional']}), of eigenvectors associated with complex eigenvalues in the eGinOE and eGinUE for $N = 10$ at fixed $\tau = 0.5$. Triangular (circular) markers represent numerical results for the eGinOE (eGinUE). Left: $\mathbb{E}_N(x,y)$ associated with purely imaginary eigenvalues as a function of $y$ at $x=0$. Right: $\mathbb{E}_N(x,y)$ associated with complex eigenvalues as a function of $x$, at a fixed $y = 0.5\sqrt{N}(1 - \tau)$. Numerically, the mean self-overlap is measured using the self-overlaps associated with the $O(10^3)$ eigenvalues nearest to the appropriate $x$ and $y$ from a data set containing $O(10^9)$ samples.
• Figure 4: Mean conditional self-overlap of eigenvectors within the bulk, $\mathbb{E}_{\text{bulk}}(x,y)$, for the eGinUE and eGinOE at large $N$ for $\tau = 0.25$ (solid lines) and $\tau = 0.75$ (dashed lines). Left: $\mathbb{E}_{\text{bulk}}(x,y)$ at fixed $x=0$ plotted as a function of $y$. Right: $\mathbb{E}_{\text{bulk}}(x,y)$ at fixed $y=0.5(1 - \tau)$ and plotted as a function of $x$. In each of these plots we have compared our finite-N formulas for $\mathbb{E}_{\text{bulk}}(x,y)$ to numerical simulations (coloured markers) of eGinOE and eGinUE matrices of size $N=500$, obtained by averaging the $O(10^3)$ self-overlaps associated with complex eigenvalues closest to the chosen $x$ and $y$ from a set of $O(10^8)$ samples.
• Figure 5: Mean conditional self-overlap, $\mathbb{E}^{\text{(eGinOE,c)}}_{\text{depletion}}(\delta,\xi)$, within the depletion regime of the eGinOE at large $N$ for a range of $\tau$. Left: $\mathbb{E}^{\text{(eGinOE,c)}}_{\text{depletion}}(\delta,\xi)$ measured as a function of $\xi$ at fixed $\delta=0$. Right: $\mathbb{E}^{\text{(eGinOE,c)}}_{\text{depletion}}(\delta,\xi)$ measured as a function of $\delta$ at fixed $\xi=0.5$. In both plots, theoretical predictions (lines) are compared to numerical simulations of $N=500$ eGinOE matrices (coloured markers). In simulations, for a required set of $\delta$ and $\xi$, mean self-overlap values are obtained by averaging self-overlaps associated with complex eigenvalues within a tolerance of $\pm 1/\sqrt{N}$ from a set containing $O(10^8)$ samples.

Theorems & Definitions (33)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Remark 2.4
• Theorem 2.5
• Theorem 2.6
• Corollary 2.7
• Remark 2.8
• Corollary 2.9
• Remark 2.10