Limit Theorems For Non-Hermitian Ensembles

TL;DR

This work derives sharp limit laws for the extreme moduli of eigenvalues in non-Hermitian random-matrix ensembles. Using Andréief integration and edge-scaling techniques, it establishes that the scaled spectral radius and the scaled minimum modulus converge to independent extreme-value limits: a standard Gumbel distribution for the outer edge and a Gumbel-type minimum for the inner edge in the complex induced Ginibre ensemble, with corresponding Rayleigh and Weibull tails for the complex Ginibre case. A key contribution is the precise tail characterizations and the demonstration of independence of the two extrema under both fixed and proportional rectangularity, highlighting universal aspects of edge statistics in these ensembles. The results provide exact edge distributions and asymptotic independence that enrich the understanding of non-Hermitian universality and may guide applications in physics and related fields where extreme eigenvalue behavior matters.

Abstract

The distribution of the modulus of the extreme eigenvalues is investigated for the complex Ginibre and complex induced Ginibre ensembles in the limit of large dimensions of random matrices. The limiting distribution of the scaled spectral radius and the scaled minimum modulus for the complex induced Ginibre ensemble, with a proportional rectangularity index, is the Gumbel distribution. The independence of these extrema is established, at appropriate scaling, for large matrices from the complex Ginibre ensemble as well as from the complex induced Ginibre ensemble for fixed and proportional rectangularity indexes. In the limit of a large size of the complex Ginibre matrices, the left and right tail distributions of the minimum modulus are the Rayleigh and Weibull distributions, respectively. The limiting left tail distribution of the minimum modulus is the same for these non-Hermitian ensembles when the rectangularity index of the complex induced Ginibre ensemble is equal to zero. This phenomenon is also verified for the right tail distribution of this minimum.

Figures (6)

• Figure 1: (left panel) Empirical probability distribution (histogram) of the scaled spectral radius $R_{N}$ for $K=10$$000$ generated matrices from the complex induced Ginibre ensemble with $N =90$, $\alpha = \frac{1}{9}$ and a proportional rectangularity index $L = \alpha N$. The corresponding exact (analytical) probability density function $p_{R_{N}}$ (solid curve). (right panel) The analytical (exact) probability density function of scaled spectral radii $R_{N}$ for $K=10$$000$ generated $N \times N$ matrices from the complex induced Ginibre ensemble with $N = 90$. The rectangularity index $L$ is proportional to $N$ such that $L = \alpha N$ with $\alpha > 0$. The results are presented for different values of $\alpha = \lbrace \frac{1}{90} , \frac{1}{9}, \frac{4}{9} \rbrace$. Graphs generated with MATLAB. Copyright Olivia V. Auster for code and graphs.
• Figure 2: Empirical probability distribution (histogram) of the scaled spectral radius $R_{N}$ for $K=10$$000$ generated matrices from the complex induced Ginibre ensemble with $N =90$, $\alpha = \frac{1}{9}$ and a proportional rectangularity index $L = \alpha N$. The exact (analytical) probability density function $p_{R_{N}}$ (blue curve). Limiting probability distributions of the scaled spectral radius $R_{N}$ presented with the red and black curves for large $N = 10e3$ and $N = 2 \times 10e3$, respectively. Graphs generated with MATLAB. Copyright Olivia V. Auster for code and graphs.
• Figure 3: (left panel) The exact cdf of the scaled spectral radius $R_{N}$ for $N = 10e3$ (blue), $N = 10e4$ (blue dotted) and $N = 10e7$ (black) and $\alpha = 1$. The asymptotic cdf of $R_{N}$ with asymptotic location and scale parameters $\mu$ and $\sigma$, respectively (red dotted curve). (right panel) The exact formulation of the scaled spectral radius $R_{N}$ cumulative distribution (black curve) for $N = 10e7$. The asymptotic cumulative distribution (red curve) parametrised with asymptotic location $\mu$ and scale $\sigma$ parameters. The parameter $\alpha = 1$ for the two curves. Graphs generated with MATLAB. Copyright Olivia V. Auster for code and graphs.
• Figure 4: (left panel) The Empirical probability distribution (histogram) of the scaled minimum modulus $r_{N}$ for $K=10$$000$ generated matrices from the complex induced Ginibre ensemble with $N =100$, $\alpha = \frac{1}{10}$ and a proportional rectangularity index $L = \alpha N, \alpha > 0$. The exact (analytical) probability density function is presented with the solid curve. (right panel) The analytical (exact) probability density function (blue curve) of the scaled minimum modulus $r_{N}$ for $K=10$$000$ generated $N \times N$ matrices from the complex induced Ginibre ensemble with $N = 100$. The rectangularity index $L$ is proportional to $N$ such that $L = \alpha N$ with $\alpha > 0$. The results are presented for different values of $\alpha = \lbrace \frac{1}{100}, \frac{1}{10}, \frac{4}{10} \rbrace$. Graphs generated with MATLAB. Copyright Olivia V. Auster for code and graphs.
• Figure 5: Empirical probability distribution (histogram) and the exact (analytical) probability density function (blue curve) of the scaled minimum modulus $r_{N}$ for $K=10$$000$ generated matrices from the complex induced Ginibre ensemble with $N =100$, $\alpha = \frac{1}{10}$ and a proportional rectangularity index $L = \alpha N$. Limiting probability distributions of the scaled minimum modulus $r_{N}$ presented with the red and black curves for large $N = 10e4$ and $N = 2 \times 10e4$, respectively. Graphs generated with MATLAB. Copyright Olivia V. Auster for code and graphs.

Theorems & Definitions (37)

• Lemma 1
• proof
• Corollary 2
• proof
• Theorem 3
• proof
• Remark 2.1
• Theorem 4
• proof
• Theorem 5