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Une brève histoire des perturbations non-hermitiennes de rang un

Guillaume Dubach, Jana Reker

TL;DR

This work surveys low-rank perturbations of random matrices in the non-Hermitian setting, focusing on rank-one perturbations across three canonical models: additive i.i.d. non-Hermitian (Ginibre) perturbations, anti-Hermitian perturbations of Hermitian matrices, and multiplicative perturbations of unitary matrices. It develops a common framework for tracking eigenvalue trajectories under perturbations, identifies conditions for the emergence of outliers, and derives precise asymptotics for their locations via tools such as the matrix determinant lemma, the resolvent (Stieltjes transform) analysis, isotropic local laws, and Rouché's theorem. The results reveal distinct outlier behaviors and scaling laws: a single outlier near the perturbation strength in the Ginibre case when vectors align, a robust outlier near $i(t-1/t)$ for the anti-Hermitian perturbation above a critical threshold, and a Gaussian-entire-function–driven Outbreak in the unitary UA model at the $N^{-1/2}$ scale, with implications for neural networks (e.g., LoRA) and open quantum systems. Collectively, these findings illuminate universal and model-specific pathways by which weak non-Hermitian perturbations generate isolated spectral features with potential practical impact in physics and machine learning.

Abstract

Les perturbations de faible rang de matrices aléatoires ont été au cœur de nombreux travaux ces vingt dernières années. En particulier, les cas non-hermitiens, moins représentés dans la littérature en règle générale, font ici l'objet d'une attention spéciale en raison de leurs applications à la physique et à l'étude des réseaux de neurones. Petit tour d'horizon. -- A brief history of non-Hermitian perturbations of rank one: Low-rank perturbations of random matrices have been the focus of active research over the past twenty years. We give an overview of different non-Hermitian models, which are generally less represented in the literature, as well as some of their applications in physics and the study of neural networks.

Une brève histoire des perturbations non-hermitiennes de rang un

TL;DR

This work surveys low-rank perturbations of random matrices in the non-Hermitian setting, focusing on rank-one perturbations across three canonical models: additive i.i.d. non-Hermitian (Ginibre) perturbations, anti-Hermitian perturbations of Hermitian matrices, and multiplicative perturbations of unitary matrices. It develops a common framework for tracking eigenvalue trajectories under perturbations, identifies conditions for the emergence of outliers, and derives precise asymptotics for their locations via tools such as the matrix determinant lemma, the resolvent (Stieltjes transform) analysis, isotropic local laws, and Rouché's theorem. The results reveal distinct outlier behaviors and scaling laws: a single outlier near the perturbation strength in the Ginibre case when vectors align, a robust outlier near for the anti-Hermitian perturbation above a critical threshold, and a Gaussian-entire-function–driven Outbreak in the unitary UA model at the scale, with implications for neural networks (e.g., LoRA) and open quantum systems. Collectively, these findings illuminate universal and model-specific pathways by which weak non-Hermitian perturbations generate isolated spectral features with potential practical impact in physics and machine learning.

Abstract

Les perturbations de faible rang de matrices aléatoires ont été au cœur de nombreux travaux ces vingt dernières années. En particulier, les cas non-hermitiens, moins représentés dans la littérature en règle générale, font ici l'objet d'une attention spéciale en raison de leurs applications à la physique et à l'étude des réseaux de neurones. Petit tour d'horizon. -- A brief history of non-Hermitian perturbations of rank one: Low-rank perturbations of random matrices have been the focus of active research over the past twenty years. We give an overview of different non-Hermitian models, which are generally less represented in the literature, as well as some of their applications in physics and the study of neural networks.
Paper Structure (4 sections, 8 theorems, 59 equations, 6 figures)

This paper contains 4 sections, 8 theorems, 59 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Le labyrinthe de Ginibre: perturbations de matrices aléatoires non-hermitiennes
  3. Les chaises musicales: perturbation anti-hermitienne d'une matrice aléatoire hermitienne
  4. Le modèle UA: perturbation multiplicative d'une matrice unitaire

Key Result

Theorem 2.1

Soit $v$ un vecteur unitaire indépendant de $G$ et $\mu$ un scalaire fixé. Alors le spectre de comporte un unique outlier $\lambda_{1} = \mu\sqrt{N}+o(1),$ tandis que le reste du spectre est contenu dans un disque $D(0,1+o(1))$ avec forte probabilité, et converge faiblement vers la loi circulaire pour $N \rightarrow + \infty$.

Figures (6)

  • Figure 1: Trajectoires de $G+tvv^{*}$ pour une matrice de Ginibre $G$ de taille $100\times 100$.
  • Figure 2: Trajectoires de $G+tvw^{*}$ pour une matrice de Ginibre $G$ de taille $100\times 100$ dans le cas où $w$ est uniforme.
  • Figure 3: Exemple d'une installation expérimentale pour observer des phénomènes chaotiques de dispersion quantique. Source: U. Kuhl.
  • Figure 4: Trajectoires des valeurs propres pour $H$ une matrice GUE de taille $100\times 100$. L'évolution du paramètre $t$ est indiquée par une variation de couleur, du noir ($t{=}0$) au bleu ($t\rightarrow\infty$).
  • Figure 5: Trajectoires du modèle $UA$ de taille $250\times 250$. L'évolution du paramètre $t$ est indiquée par la variation de couleur de noir $(|t|{=}1)$ à bleu ($t\rightarrow0$).
  • ...and 1 more figures

Theorems & Definitions (8)

  • Theorem 2.1: Tao Tao2013
  • Theorem 2.2
  • Theorem 3.1: O'Rourke et Matchett Wood ORourkeWood2017
  • Theorem 3.2: Dubach et Erdős DubachErdos
  • Theorem 3.3: Fyodorov, Khoruzhenko et Poplavskyi FyodorovGUE
  • Theorem 4.1: Fyodorov Fyodorov2001
  • Theorem 4.2: Forrester et Ipsen ForresterIpsen
  • Theorem 4.3: Dubach et Reker DubachReker