Rectangular Complex Ginibre Matrices

• Rectangular complex Ginibre matrices are non-Hermitian random matrices defined by independent standard complex Gaussian entries, with unequal dimensions influencing spectral behavior.
• The induced Ginibre ensemble method quadratises rectangular matrices to reveal eigenvalue repulsion, annular spectral support, and detailed extreme value statistics.
• Determinantal point processes and contour integral analyses uncover universal bulk and edge scaling laws, underpinning significant applications in physics and quantum information.

Rectangular complex Ginibre matrices are non-Hermitian random matrices with independently distributed complex Gaussian entries, where the number of rows $M$ and columns $N$ need not be equal. The paper of these ensembles is central to random matrix theory and has significant applications in physics, quantum information, statistical mechanics, and beyond. Their spectral properties, particularly for ensembles formed via matrix products or as the so-called induced Ginibre ensembles, reveal a profound interplay between global density, local spectral universality, extreme value statistics, integrable structure, and eigenvector correlations.

1. Definition, Construction, and Induction Procedures

A rectangular complex Ginibre matrix $X$ of size $M \times N$ has entries drawn independently from standard complex normal distributions. The statistical properties of its singular values and (for appropriately defined ensemble extensions) its eigenvalues have deep implications for non-Hermitian random matrix theory.

The induced Ginibre ensemble (sometimes called the quadratised Ginibre ensemble) generalizes the standard Ginibre ensemble to allow rectangularity to be encoded as a parameter $L \geq 0$:

$$p_{\operatorname{IndG}}(G) \propto (\det G^\dagger G)^{L} e^{-\operatorname{Tr} G^\dagger G},\quad G \in \mathbb{C}^{N \times N}$$

where $G$ is obtained by the "quadratisation" of a rectangular $M \times N$ Ginibre matrix $X$ with $M = N + L$, typically via a block-unitary transformation $$W^\dagger X = \begin{pmatrix} G \ 0 \end{pmatrix}$$ for $W \in U(M)$ (Fischmann et al., 2011).

The motivation for "quadratisation" is that for many physical and quantum information problems only spectral data of some square contraction (or extension) of the physically relevant rectangular matrix is meaningful.

2. Joint Probability Densities and the Role of Rectangularity

The joint eigenvalue probability density (jpdf) of the induced Ginibre ensemble becomes

$$P(\lambda_1, \dots, \lambda_N) = \frac{1}{N! \pi^N} \prod_{j=1}^N \frac{1}{\Gamma(j+L)} \prod_{j<k}|\lambda_j - \lambda_k|^2 \prod_{j=1}^N |\lambda_j|^{2L} e^{-|\lambda_j|^2}$$

so that the determinant power $|\det G|^{2L}$ induces a radial repulsion of eigenvalues from the origin. This leads in the large-$N$ limit to eigenvalues being supported uniformly on an annulus with radii

$$r_{\text{in}} = \sqrt{L},\qquad r_{\text{out}} = \sqrt{N+L}$$

with uniform bulk density and universal edge behavior, described by error functions at the boundaries (Fischmann et al., 2011, Auster, 7 May 2025). For $L=0$, the standard Ginibre disk is recovered; for $L \propto N$ (rectangularity proportional to dimension), the spectrum is uniformly supported on a nontrivial annulus.

In the setting of products of $M$ rectangular Ginibre matrices with consistent rectangularity, the squared singular values have a global density with support and functional form governed by the rectangularity parameter $y$,

$y = \lim_{N\to\infty} \frac{N}{N_\ell}\in(0,1]$

for each factor $X_\ell$ (of size $N_\ell\times N_{\ell-1}$), with the explicit spectral density described parametrically via trigonometric functions of auxiliary variables as derived from the equation for the Stieltjes transform (Gu, 20 Oct 2025, Kuijlaars et al., 2013).

3. Correlation Kernels, Universal Scaling, and Integrable Structures

Rectangular complex Ginibre ensembles exhibit determinantal point process structure in both their eigenvalues and singular values (Kuijlaars et al., 2013, Liu et al., 2014). For the squared singular values of a product ensemble, the corresponding correlation kernel can be written (expressed for $M$ factors) as a double contour integral involving Meijer $G$-functions:

$$K_n(x,y) = \frac{1}{(2\pi i)^2} \int_{-1/2-i\infty}^{-1/2+i\infty} ds \oint_\Sigma dt\, \cdots \frac{x^t y^{-s-1}}{s-t}$$

with parameters encoding the rectangularity of each matrix factor (Kuijlaars et al., 2013, Liu et al., 2014).

Rigorous asymptotic analysis in the bulk and at the spectral edges yields universal scaling limits:

• In the bulk, the sine kernel emerges for appropriately rescaled local statistics of squared singular values,

$$K_{\sin}(\xi,\eta) = \frac{\sin\pi(\xi-\eta)}{\pi(\xi-\eta)}$$

irrespective of the global density or precise rectangularity (Liu et al., 2014, Gu, 20 Oct 2025).

• At the soft edge, the Airy kernel becomes universal in the high-depth/low-width regime for products, while a Gaussian fluctuation regime can occur for high depth-to-width ratio (DWR), with a critical kernel interpolating between these as DWR varies (Gu, 10 Jul 2025).

At the hard edge (vicinity of the origin), the limiting kernels are expressed via Meijer $G$-functions (or, equivalently, hyper-Bessel functions), generalizing the Bessel kernel. For $M=2$ products, these coincide with analogous kernels from the Cauchy-Laguerre two-matrix model, providing evidence for a universality class specific to products of random matrices (Kuijlaars et al., 2013). For $M=1$ (square or rectangular), the hard-edge gap probability is governed by a Painlevé III' equation; for $M=2$ and general rectangularity, a fourth-order nonlinear equation controls the gap probability, with reduction to lower-order equations in special cases (Witte et al., 2016).

4. Extreme Value and Edge Statistics

Analysis of the norm and minimum modulus of eigenvalues in (rectangular) Ginibre ensembles yields exact limiting distributions:

• The spectral radius (after proper centering and scaling) follows a Gumbel (maximum) distribution,

$$\mathbb{P}\left(\frac{r_{\text{max}}^{(N)}(G)}{\sqrt{(1+\alpha)N}} \leq ... \right) \to \exp(-\exp(-x))$$

• The minimum modulus (similarly rescaled near the inner edge) converges to a Gumbel (minimum) law (Auster, 7 May 2025).
• These extremes are asymptotically independent for both standard and induced Ginibre ensembles at large $N$.
• The joint law of the moduli (for $L=0$) coincides with independent Gamma-Rayleigh random variables; for the minimum modulus, the left tail is Rayleigh and the right tail is Weibull, with the Weibull exponent reflecting the rectangularity index $L$.

This framework covers the transition of the origin from a soft edge ($L$ small) to a strict hard edge ($L$ large), and underscores the distinctive "annular" single-ring support for large proportional rectangularity.

5. Bulk and Microscopic Universality

Despite dramatic differences in global density between rectangular and square Ginibre ensembles (disk versus annulus; power-law near the edges versus hard edges), local statistics in both bulk and at soft/hard edges are robust:

• Bulk correlations of singular values and eigenvalues are universally described by the sine kernel, regardless of product depth or rectangularity, provided the matrices are sufficiently large and the scaling limits are well-chosen (Gu, 20 Oct 2025, Liu et al., 2014, Kuijlaars et al., 2013).
• Edge regimes exhibit phase transitions in fluctuations. For products of depth $M$ and width $N$, the depth-to-width ratio

$\Delta_{M,N} = \sum_{j=0}^M \frac{1}{N+v_j}$

acts as a critical parameter: for $\Delta_{M,N}\to\infty$ the edge behaves Gaussian (central limit scaling), while $\Delta_{M,N}\to 0$ yields Airy kernel fluctuations. The critical kernel connects these regimes (Gu, 10 Jul 2025).

These universality features extend to more general induced or biorthogonal ensembles, revealing that spectral repulsion and scaling statistics are insensitive to matrix aspect ratios or the precise structure of the product.

6. Lyapunov Exponents for Products

Consider the Lyapunov exponents of products of $T$ independent rectangular complex Ginibre matrices (or mixed products). As $T,n\to\infty$,

• The exponents are asymptotically equally spaced ("picket fence" statistics),
• The individual exponents are given approximately by harmonic numbers (for the Ginibre case): $\lambda_i(n) \approx H_{n-i} - \gamma$, with $H_m$ the $m$-th harmonic number and $\gamma$ the Euler-Mascheroni constant,
• The joint fluctuations for the exponents approach deterministic limits, with the moduli of eigenvalues becoming independent and log-normally distributed in the large-pro-duct limit (Ipsen, 2014, Ahn et al., 2021).

This behavior is explained via connections to multiplicative Brownian motion on $\operatorname{GL}_n(\mathbb{C})$, and is underpinned by contour integral techniques for moments of singular values.

7. Simulation, Numerical Range, and Linear Statistics

Simulation of the eigenvalue process for (rectangular) Ginibre ensembles leverages truncated kernels and projection onto compact subsets for efficiency. Sampling the eigenvalues of random matrices with complex Gaussian entries through eigenvalue decomposition yields highly effective simulations, extendable to the rectangular and induced ensemble cases (Decreusefond et al., 2013).

For the square Ginibre, the numerical range converges to a disk of radius $\sqrt{2}$, exceeding the support of the eigenvalues by a belt of width $\sqrt{2}-1$—a geometric quantification of non-normality. The operator norm for upper-triangular Ginibre matrices converges to $\sqrt{2e}$ (Collins et al., 2013). For rectangular matrices, similar metrics may be defined in terms of the singular value support.

Linear statistics (i.e., fluctuations of spectral measures for smooth test functions) obey central limit theorems where variance splits into $H^1$ and $H^{1/2}$ contributions (for bulk and edge), with universality holding whenever entry distributions match the first four moments of the complex normal distribution (Kopel, 2015).

---

These structural, spectral, and statistical features of rectangular complex Ginibre matrices not only generalize classical results from random matrix theory but also cement the induced, product, and rectangular Ginibre ensembles as paradigmatic in the paper of universal non-Hermitian phenomena across mathematics and physics.