The delocalization of eigenvectors of real elliptic matrices

TL;DR

The paper investigates how eigenvector localization in real orthogonally invariant random matrices depends on eigenvalue location. It develops a precise, Schur-decomposition-based framework for the real Elliptic Ginibre ensemble, linking the inverse participation ratio (IPR) of an eigenvector to a local 2×2 block and Legendre polynomials, yielding exact and limiting distributions. Three regimes emerge: real-axis eigenvalues produce maximal localization with $\mathrm{IPR}_q\to(2q-1)!!$, non-real eigenvalues in the bulk yield $\mathrm{IPR}_q\to q!$, and a depletion regime with $\Im \lambda = y/\sqrt{N}$ yields a continuous family $\ell_{q,y}=q!S_y^{-q}L_q(S_y)$ whose distribution depends only on $y$. This interpolation explains the observed increased localization near the real axis and suggests universality across a broad class of orthogonally invariant ensembles through the local Schur-block analysis. The results provide explicit density formulas for the depletion parameter and establish concentration and almost-sure convergence, highlighting the role of the local 2×2 block structure in determining eigenvector localization in non-Hermitian real random matrices.

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 $λ$ with $|\mathfrak{Im}(λ)| = y / \sqrt{N}$, 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 $τ\in [0,1[$, including the classical real Ginibre ensemble ($τ=0$).

Figures (4)

• Figure 1: Eigenvalues of 10 realizations of $N\times N$ matrices ($N=2000$) with real entries (left) and complex entries (right). The color of each eigenvalue represents its inverse participation ratio, a measure of its delocalization. For the Real Ginibre ensemble, eigenvectors with eigenvalues close to $\mathbb{R}$ are visibly more localized than the others.
• Figure 2: Some densities $\gamma_{2,y}$ from Corollary \ref{['cor:smalllq']} for various $y$ ranging from $0$ (violet) to $2$ (yellow). When $y$ is large, the distribution converges towards 2, while for small $y$ it converges towards 3.
• Figure 3: Three spectra from the real Elliptic Ginibre ensemble at different hermiticity levels $\tau$. The colorbar is the same as in Figure \ref{['fig:three graphs']}.
• Figure 4: Spectra of other models of non-Hermitian random matrices. The scale for the IPRs is exactly the same as in \ref{['fig:rge']} (violet is 2 and yellow is 3 or more). Spectra are normalized to be in $D(0,1)$.

Theorems & Definitions (24)

• Lemma 2.1
• proof
• Theorem 2.2
• Corollary 2.3
• Lemma 5.1
• proof
• Proposition 6.1
• proof
• Lemma 6.2
• proof