The delocalization of eigenvectors of real elliptic matrices

TL;DR

This paper studies how eigenvector localization varies in real elliptic matrices, revealing that localization depends on the eigenvalue's proximity to the real axis and providing explicit distributional results.

Contribution

It introduces a detailed analysis of eigenvector delocalization in real elliptic matrices, including explicit distributional limits conditioned on eigenvalue location.

Findings

Eigenvector IPR converges to a distribution depending on eigenvalue imaginary part.

Localization increases as eigenvalues approach the real axis.

Results apply to various ensembles, including the real Ginibre ensemble.

Abstract

We investigate delocalization phenomena for eigenvectors of real random matrices that are invariant by orthogonal transformations. A specific phenomenon with these ensembles is that an eigenvector is typically more localized when its eigenvalue is closer to the real axis while for unitarily invariant ensembles, all eigenvectors are delocalized at the same level. More precisely, we measure the delocalization level of a vector $x\in \mathbb{C}^N$ using the Inverse Participation Ratio $\mathrm{IPR}(x) = N|x|_4^4 / |x|_2^4 \geqslant 1$. A higher IPR means a more localized vector. Using the exact distribution of the Schur decomposition of some paradigmatic rotation-invariant matrix models, we prove that conditionally on having an eigenvalue $\lambda$ with $|\mathfrak{Im}(\lambda)| = y / \sqrt{N}$, the IPR of the associated eigenvector converges in distribution towards a random variable $\ell_y$ with an explicit density depending only on $y$. We then prove that $\ell_y \to 3$ when $y \to 0$ and $\ell_y \to 2$ when $y\to +\infty$, coherently with the observed phenomenon. This result is explicitly proved for higher-order IPRs and for the real Elliptic Ginibre ensemble at every non-symmetry parameter $\tau \in [0,1[$, including the classical real Ginibre ensemble ($\tau=0$).