Optimal decay of eigenvector overlap for non-Hermitian random matrices

TL;DR

This work establishes that the standard eigenvector overlaps for large non-Hermitian i.i.d. random matrices decay quadratically with the eigenvalue separation, uniformly across the spectrum. The authors develop a two-resolvent local law for the Hermitisation $H^z$, and integrate a multi-resolvent framework with a zig flow (propagating bounds via an Ornstein--Uhlenbeck perturbation) and a zag step (Green function comparison) to control products of resolvents, including near the spectral edge. The main contributions are (i) a uniform, high-probability bound on off-diagonal overlaps $\mathcal{O}_{ij}$ and their normalized forms, (ii) a robust method that extends results known for complex Ginibre to general i.i.d. ensembles in both symmetry classes, and (iii) refined edge-local laws that yield sharp $|z_1-z_2|^{-2}$ scaling even at the spectral edge. Overall, the paper advances understanding of eigenvector overlaps in non-Hermitian random matrices and provides powerful multi-resolvent tools for studying non-normal spectral data with broad applicability. $

Abstract

We consider the standard overlap $\mathcal{O}_{ij}: =\langle \mathbf{r}_j, \mathbf{r}_i\rangle\langle \mathbf{l}_j, \mathbf{l}_i\rangle$ of any bi-orthogonal family of left and right eigenvectors of a large random matrix $X$ with centred i.i.d. entries and we prove that it decays as an inverse second power of the distance between the corresponding eigenvalues. This extends similar results for the complex Gaussian ensemble from Bourgade and Dubach [arXiv:1801.01219], as well as Benaych-Georges and Zeitouni [arXiv:1806.06806], to any i.i.d. matrix ensemble in both symmetry classes. As a main tool, we prove a two-resolvent local law for the Hermitisation of $X$ uniformly in the spectrum with optimal decay rate and optimal dependence on the density near the spectral edge.

Figures (2)

• Figure 1: The shaded square indicates the region in the $(\eta,t)$ plane where the local law holds with an error bound $\mathcal{E}_b$, as given in assumption (\ref{['assump_R']}). Starting from the global law at the initial black dot ($\eta_*^{(0)} \sim 1)$, the slanted line downward indicates the zig step using Proposition \ref{['prop:isoflow']}, while the horizontal line leftward indicates the zag step using Lemma \ref{['lemma_RS_gft']} and a Gronwall argument. Iterating finitely many zig-zags until reaching the final red point ($\eta_*^{(K)} \sim \eta_*$, $K=O(\epsilon^{-1})$) yields the improved local law with an improved error $\mathcal{E}_{b+\theta}$ for any $\min_{i=1}^{2}|\eta_i| \rho_i \gtrsim n^{-1+\epsilon}$.
• Figure 2: In each step we use the same zig-zag as explained in Figure \ref{['fig:zigzag']} to improve the local law obtained in the previous step (with the error bound $\mathcal{E}_{(k-1)\theta}$ indicated on the upper right corner of the shaded square) to the next order bound $\mathcal{E}_{k\theta}$ (indicated along the zigzag lines). Starting from Theorem \ref{['prop_initial']} as the initial input, we iterate this process for $\lceil \frac{1}{\theta} \rceil$ times to improve the a priori error bound $\mathcal{E}_0$ to the desired $\mathcal{E}_1$.

Theorems & Definitions (40)

• Theorem 2.2
• Remark 2.3
• Remark 2.4
• Corollary 2.5
• Remark 2.6
• Theorem 3.1
• Corollary 3.2
• Lemma 3.3
• Theorem 3.4
• Theorem 3.5