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Optimal Lower Bound on Eigenvector Overlaps for non-Hermitian Random Matrices

Giorgio Cipolloni, László Erdős, Joscha Henheik, Dominik Schröder

TL;DR

The paper addresses eigenvector stability and singular-vector thermalisation for large non-Hermitian random matrices with additive i.i.d. noise. It develops a novel observables decomposition into regular and singular parts and establishes two-resolvent local laws for regular observables, enabling a rigorous ETH-like statement for the Hermitised matrix and an optimal lower bound on diagonal eigenvector overlaps. The key technical advance is the structured regularisation anchored by two critical directions, plus a comprehensive hierarchy of master and reduction inequalities that control long resolvent chains. Collectively, these results extend QUE from Hermitian Wigner ensembles to general non-Hermitian models with i.i.d. perturbations, with implications for eigenvalue sensitivity and Dyson-type dynamics in the non-Hermitian setting, and provide sharp quantitative control over eigenvector overlaps beyond the Ginibre case.

Abstract

We consider large non-Hermitian $N\times N$ matrices with an additive independent, identically distributed (i.i.d.) noise for each matrix elements. We show that already a small noise of variance $1/N$ completely thermalises the bulk singular vectors, in particular they satisfy the strong form of Quantum Unique Ergodicity (QUE) with an optimal speed of convergence. In physics terms, we thus extend the Eigenstate Thermalisation Hypothesis, formulated originally by [Deutsch 1991] and proven for Wigner matrices in [Cipolloni, Erdős, Schröder 2020], to arbitrary non-Hermitian matrices with an i.i.d. noise. As a consequence we obtain an optimal lower bound on the diagonal overlaps of the corresponding non-Hermitian eigenvectors. This quantity, also known as the (square of the) eigenvalue condition number measuring the sensitivity of the eigenvalue to small perturbations, has notoriously escaped rigorous treatment beyond the explicitly computable Ginibre ensemble apart from the very recent upper bounds given in [arXiv:2005.08930] and [arXiv:2005.08908]. As a key tool, we develop a new systematic decomposition of general observables in random matrix theory that governs the size of products of resolvents with deterministic matrices in between.

Optimal Lower Bound on Eigenvector Overlaps for non-Hermitian Random Matrices

TL;DR

The paper addresses eigenvector stability and singular-vector thermalisation for large non-Hermitian random matrices with additive i.i.d. noise. It develops a novel observables decomposition into regular and singular parts and establishes two-resolvent local laws for regular observables, enabling a rigorous ETH-like statement for the Hermitised matrix and an optimal lower bound on diagonal eigenvector overlaps. The key technical advance is the structured regularisation anchored by two critical directions, plus a comprehensive hierarchy of master and reduction inequalities that control long resolvent chains. Collectively, these results extend QUE from Hermitian Wigner ensembles to general non-Hermitian models with i.i.d. perturbations, with implications for eigenvalue sensitivity and Dyson-type dynamics in the non-Hermitian setting, and provide sharp quantitative control over eigenvector overlaps beyond the Ginibre case.

Abstract

We consider large non-Hermitian matrices with an additive independent, identically distributed (i.i.d.) noise for each matrix elements. We show that already a small noise of variance completely thermalises the bulk singular vectors, in particular they satisfy the strong form of Quantum Unique Ergodicity (QUE) with an optimal speed of convergence. In physics terms, we thus extend the Eigenstate Thermalisation Hypothesis, formulated originally by [Deutsch 1991] and proven for Wigner matrices in [Cipolloni, Erdős, Schröder 2020], to arbitrary non-Hermitian matrices with an i.i.d. noise. As a consequence we obtain an optimal lower bound on the diagonal overlaps of the corresponding non-Hermitian eigenvectors. This quantity, also known as the (square of the) eigenvalue condition number measuring the sensitivity of the eigenvalue to small perturbations, has notoriously escaped rigorous treatment beyond the explicitly computable Ginibre ensemble apart from the very recent upper bounds given in [arXiv:2005.08930] and [arXiv:2005.08908]. As a key tool, we develop a new systematic decomposition of general observables in random matrix theory that governs the size of products of resolvents with deterministic matrices in between.
Paper Structure (36 sections, 34 theorems, 460 equations, 4 figures)

This paper contains 36 sections, 34 theorems, 460 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Non-Hermitian eigenvector overlaps
  3. Thermalisation of singular vectors
  4. Structural decomposition of observables
  5. Notations
  6. Main results
  7. Non-Hermitian singular vectors and eigenvectors
  8. Eigenstate Thermalisation Hypothesis for the Hermitisation of $X+{\Lambda}$
  9. Proof of the main results
  10. Regular observables: A bound on \ref{['eq:2Gav']}
  11. Estimating \ref{['eq:2Gav']} for general observables
  12. Proof of the main results
  13. Proof of Theorem \ref{['thm:main']}
  14. Proof of Theorem \ref{['thm:main singvec']}
  15. Proof of Theorem \ref{['thm:overlap']}
  16. ...and 21 more sections

Key Result

Theorem 2.2

(Thermalisation of Singular Vectors) Fix a bounded ${\Lambda} \in \mathbf{C}^{N \times N}$ and $\kappa > 0$ independent of $N$. Let $\{\boldsymbol{u}_i\}_{i \in [N]}$ and $\{\boldsymbol{v}_i\}_{i \in [N]}$ be the (normalised) left- and right-singular vectors of $X+{\Lambda}$, respectively, where $X$ where the maximum is taken over all $i,j \in [N]$ such that the quantiles $\gamma_i, \gamma_j \in \

Figures (4)

  • Figure 1: Depicted is the density $\rho$ for the deformation ${\Lambda} = - z$ with $|z|$ slightly less than one. On the horizontal axis, we indicated the two components of the bulk $\mathbf{B}_\kappa$. The distance between $\mathbf{B}_\kappa$ and a regular edge scales like $\kappa^{2/3}$, while near an (approximate) cusp the distance between the two components scales linearly (see also \ref{['eq:bulk']} and \ref{['kappabulkreg']}).
  • Figure 2: Depicted are the target spectral domain \ref{['eq:Omega']}, the initial spectral domain \ref{['eq:Omega00']} and four intermediate domains from the family \ref{['eq:Omegal']}. The solid black curve represents the symmetric scDos $\rho$ for the perturbation ${\Lambda} = -z$ with $|z|$ slightly less than one (see Example \ref{['exam:Defo=-z']}). Close to a regular edge of the scDos, the horizontal distance between two domains scales like $\kappa^{2/3}$. Near an (approximate) cusp, the scaling agrees with the linear lower bound given in \ref{['kappabulkreg']}.
  • Figure 3: Depicted is the scenario from Lemma \ref{['lem:intrepG^2']} with five spectral parameters represented as dots in the upper half plane. Moreover, we indicated the union of compact intervals $J$ on the real axis and the contour $\Gamma$ as described in \ref{['eq:contour']}. Note that one of the three intervals constituting $J$ does not contain any $\mathrm{Re}\, w_j$.
  • Figure 4: The contour $\Gamma$ is split into three parts (see \ref{['eq:contourdecomp']}). In case of multiple spectral parameters, the second part might require a further decomposition at the level indicated by the dashed horizontal line (see Footnote \ref{['ftn:furthersplit']}). Depicted is the situation, where the bulk $\mathbf{B}_{\ell \kappa_0}$ consists of two components.

Theorems & Definitions (76)

  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • Example 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Remark 2.8
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • ...and 66 more