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Global and local limits for products of rectangular Ginibre matrices

Yandong Gu

TL;DR

This work studies singular value statistics for products of independent rectangular complex Ginibre matrices in a regime where the depth-to-width parameter vanishes and the rectangularity ratios converge to constants. A two-step approach derives the global limiting density for squared singular values via a Stieltjes transform $G(z)$ solving $1 - zG(z) + G(z)\prod_{l=1}^{M}(1 - y_l + z y_l G(z)) = 0$, followed by an analysis showing that, after microscopic scaling, the local bulk correlations converge to the universal sine kernel $K_{\text{sin}}(\xi_i,\xi_j)$. In the homogeneous case $y_l=y$, the authors obtain explicit parametric representations for the density and spectral edges, with the bulk statistics proven to be sine-kernel universal for $1 \ll M \ll N$ as $\Delta_{M,N}\to 0$, extending Fuss–Catalan-type results from square to rectangular matrix products. The results offer a framework for understanding global and local spectral behavior in rectangular random matrix products and have potential applications in stability analysis for deep neural networks and communication systems.

Abstract

We investigate singular value statistics for products of independent rectangular complex Ginibre matrices. When the rectangularity parameters of the matrices converge to a common limit in the asymptotic regime, the limiting spectral density is derived, and the local statistics in the bulk are shown to be governed by the universal sine kernel. This generalizes the classical results for products of square Ginibre matrices to a specific class of rectangular matrix products.

Global and local limits for products of rectangular Ginibre matrices

TL;DR

This work studies singular value statistics for products of independent rectangular complex Ginibre matrices in a regime where the depth-to-width parameter vanishes and the rectangularity ratios converge to constants. A two-step approach derives the global limiting density for squared singular values via a Stieltjes transform solving , followed by an analysis showing that, after microscopic scaling, the local bulk correlations converge to the universal sine kernel . In the homogeneous case , the authors obtain explicit parametric representations for the density and spectral edges, with the bulk statistics proven to be sine-kernel universal for as , extending Fuss–Catalan-type results from square to rectangular matrix products. The results offer a framework for understanding global and local spectral behavior in rectangular random matrix products and have potential applications in stability analysis for deep neural networks and communication systems.

Abstract

We investigate singular value statistics for products of independent rectangular complex Ginibre matrices. When the rectangularity parameters of the matrices converge to a common limit in the asymptotic regime, the limiting spectral density is derived, and the local statistics in the bulk are shown to be governed by the universal sine kernel. This generalizes the classical results for products of square Ginibre matrices to a specific class of rectangular matrix products.
Paper Structure (5 sections, 3 theorems, 59 equations, 1 figure)

This paper contains 5 sections, 3 theorems, 59 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Main results
  4. Proof of Theorem \ref{['bulkthm']}
  5. Concluding remarks

Key Result

Theorem 1.1

Assume $\lim_{N \to \infty} \Delta_{M,N} = 0$. For $\theta \in (0, \pi)$, let and the scaling factor is where $x(\theta)$ is defined in x_theta and $r$ is given by r_para. The following limits for correlation functions of eigenvalues of $\log(Y^*_M Y_M)$ hold uniformly for $\xi_1,\dots,\xi_n$ in any compact subset of $\mathbb{R}$.

Figures (1)

  • Figure 1: Schematic contours in the proof of Theorem \ref{['bulkthm']}

Theorems & Definitions (7)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.1
  • proof : Proof of Theorem \ref{['bulkthm']}
  • Lemma 2.2
  • proof : Proof of Lemma \ref{['glolemma']}
  • proof : Proof of Lemma \ref{['lolemma']}