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Real Eigenvalues of Asymmetric Wishart Matrices: Expected Number, Global Density and Integrable Structure

Sung-Soo Byun, Kohei Noda

TL;DR

This work analyzes the real eigenvalues of asymmetric Wishart matrices X = X_+ X_-^T with X_± built from rectangular Gaussian matrices, introducing a Pfaffian real-eigenvalue point process and an integrable decomposition that links the real-kernel to the complex Laguerre kernel. The authors derive explicit leading-order counts for real eigenvalues in both strongly (E_{N,τ}^ν ∼ c(τ,ρ) N^{1/2}) and weakly (E_{N,τ}^ν ∼ c(α) N) non-Hermitian regimes, and they obtain macroscopic limiting densities that interpolate between square-root edge behavior and the Marchenko–Pastur law as τ → 1. A central contribution is a decomposition of the real-kernel into components expressed via the complex counterpart, enabling a transfer of analytic machinery from complex ensembles to the real case. The results illuminate universality across non-Hermitian random-matrix families and provide precise asymptotics that connect to known symmetric cases and elliptic Ginibre models. The methods, rooted in an integrable structure with Laguerre polynomials, may extend to broader real asymmetric ensembles and offer tools for studying finite-N corrections and fluctuations of real eigenvalues.

Abstract

We investigate the real eigenvalues of asymmetric Wishart matrices of size $N$, indexed by the rectangular parameter $ν\in \mathbb{N}$ and the non-Hermiticity parameter $τ\in [0,1]$. The rectangular parameter $ν$ is either fixed or proportional to $N$. The non-Hermiticity parameter $τ$ is either fixed or $τ= 1 - O(1/N)$, corresponding to the strongly and weakly non-Hermitian regimes, respectively. We establish a decomposition structure for the finite-$N$ correlation kernel of the real eigenvalues, which form Pfaffian point processes. Taking the symmetric limit $τ= 1$, where the model reduces to the Laguerre orthogonal ensemble, this decomposition structure reduces to the known rank-one perturbation structure established by Adler, Forrester, Nagao, and van Moerbeke, as well as by Widom. Using the decomposition structure, we show that the expected number of real eigenvalues is proportional to $\sqrt{N}$ in the strongly non-Hermitian regime and to $N$ in the weakly non-Hermitian regime, providing explicit leading coefficients in both cases. Furthermore, we derive the limiting real eigenvalue densities, which recovers the Marchenko-Pastur distribution in the symmetric limit.

Real Eigenvalues of Asymmetric Wishart Matrices: Expected Number, Global Density and Integrable Structure

TL;DR

This work analyzes the real eigenvalues of asymmetric Wishart matrices X = X_+ X_-^T with X_± built from rectangular Gaussian matrices, introducing a Pfaffian real-eigenvalue point process and an integrable decomposition that links the real-kernel to the complex Laguerre kernel. The authors derive explicit leading-order counts for real eigenvalues in both strongly (E_{N,τ}^ν ∼ c(τ,ρ) N^{1/2}) and weakly (E_{N,τ}^ν ∼ c(α) N) non-Hermitian regimes, and they obtain macroscopic limiting densities that interpolate between square-root edge behavior and the Marchenko–Pastur law as τ → 1. A central contribution is a decomposition of the real-kernel into components expressed via the complex counterpart, enabling a transfer of analytic machinery from complex ensembles to the real case. The results illuminate universality across non-Hermitian random-matrix families and provide precise asymptotics that connect to known symmetric cases and elliptic Ginibre models. The methods, rooted in an integrable structure with Laguerre polynomials, may extend to broader real asymmetric ensembles and offer tools for studying finite-N corrections and fluctuations of real eigenvalues.

Abstract

We investigate the real eigenvalues of asymmetric Wishart matrices of size , indexed by the rectangular parameter and the non-Hermiticity parameter . The rectangular parameter is either fixed or proportional to . The non-Hermiticity parameter is either fixed or , corresponding to the strongly and weakly non-Hermitian regimes, respectively. We establish a decomposition structure for the finite- correlation kernel of the real eigenvalues, which form Pfaffian point processes. Taking the symmetric limit , where the model reduces to the Laguerre orthogonal ensemble, this decomposition structure reduces to the known rank-one perturbation structure established by Adler, Forrester, Nagao, and van Moerbeke, as well as by Widom. Using the decomposition structure, we show that the expected number of real eigenvalues is proportional to in the strongly non-Hermitian regime and to in the weakly non-Hermitian regime, providing explicit leading coefficients in both cases. Furthermore, we derive the limiting real eigenvalue densities, which recovers the Marchenko-Pastur distribution in the symmetric limit.

Paper Structure

This paper contains 18 sections, 27 theorems, 268 equations, 5 figures.

Table of Contents

  1. Introduction and main results
  2. Integrable structure of real eigenvalues
  3. Preliminary analysis for the correlation kernel
  4. Plancherel-Rotach asymptotis of the Laguerre polynomials
  5. Complex correlation kernel and rescaling
  6. Differential equation for complex counterpart of kernel
  7. Auxiliary asymptotics
  8. Proofs of Theorems \ref{['Thm_expected number']} and \ref{['Thm_global density']} at strong non-Hermiticity
  9. Outline of the proof
  10. Macroscopic behaviour of the one-point density; Proof of Proposition \ref{['Prop_asymptotic of 1pt function RN strong']}
  11. L1-estimate for linear statistics; Proof of Lemma \ref{['lem L1 estimate']}
  12. Proofs of Theorems \ref{['Thm_expected number']} and \ref{['Thm_global density']} at weak non-Hermiticity
  13. Outline of the proof
  14. Macroscopic behaviour of the one-point density; Proof of Proposition \ref{['prop of one density weak macro']}
  15. L1-estimate for linear statistics; Proof of Lemma \ref{['Lem L1 estimate at weak']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

Suppose that $\nu = \varrho N$ with $\varrho > 0$ fixed, or that $\nu \geq 0$ is fixed.

Figures (5)

  • Figure 1: The plots show the eigenvalues of asymmetric Wishart matrices with $N=400$ and $\tau=1/2$. In the zoomed-in images, the presence of purely real eigenvalues is clearly visible. Furthermore, in the case of $\nu=0$, there is an accumulation of eigenvalues near the origin, reflecting the divergent density.
  • Figure 2: Plot (A) shows $\tau \mapsto E_{N,\tau}^\nu / \sqrt{N}$ (blue dots) and its comparison with $c(\tau, \varrho)$ (solid gray line). Plot (B) illustrates $\alpha \mapsto E_{N,\tau}^\nu / N$ (blue dots) for $\tau$ given by \ref{['tau AH']}, compared to $c(\alpha)$ (solid gray line). Here, we use 20 samples of $X$ with size $N = 400$ and $\nu = N$.
  • Figure 3: The plots (A)--(C) present histograms of real eigenvalues for 20 samples of $X$ with size $N=2000$, compared to $\rho_{\tau,\varrho}^s$, where $\tau = 1/\sqrt{2}$. Similarly, the plots (D)--(F) illustrate histograms of real eigenvalues for 20 samples of $X$ with size $N=400$, where $\tau$ is defined by \ref{['tau AH']}, compared to $\rho_{\alpha,\varrho}^w$, with $\alpha = 1$.
  • Figure 4: The plots the boundary of the droplet $S_\varrho$ in \ref{['droplet']}, along with the right and left endpoints $\xi_\pm$ and the foci $F_\pm$. Here, $\varrho=3$ thereby $\tau_{\varrho}=0.5$.
  • Figure 5: Illustration of the decomposition of $I_{ \xi_+ }$

Theorems & Definitions (57)

  • Theorem 1.1: Expected number of real eigenvalues
  • Remark 1.1: Alternative expressions
  • Remark 1.2: Special case $\varrho=0$ and comparison with previous results at strong non-Hermiticity
  • Remark 1.3: Independence of the rectangular parameter at weak non-Hermiticity
  • Theorem 1.2: Limiting density of real eigenvalues
  • Remark 1.4: Normalisation constants
  • Remark 1.5: Square root relation between complex and real eigenvalue densities at strong non-Hermiticity
  • Remark 1.6: Interpolating properties
  • Remark 1.7: Comparison with the elliptic Ginibre matrix
  • Theorem 2.1: Decomposition structure of the correlation kernel
  • ...and 47 more