On delocalization of eigenvectors of random non-Hermitian matrices

TL;DR

This paper investigates the delocalization properties of eigenvectors of random non-Hermitian matrices with i.i.d entries, providing improved bounds on how spread out these vectors are across their components.

Contribution

It establishes new, sharper lower bounds on eigenvector delocalization for both real and complex i.i.d matrices, surpassing previous results and approaching optimality.

Findings

Eigenvectors of real matrices are highly delocalized with high probability.

Eigenvectors of complex matrices are also delocalized with high probability.

Bounds are nearly optimal, matching Gaussian matrix cases up to polylogarithmic factors.

Abstract

We study delocalization of null vectors and eigenvectors of random matrices with i.i.d entries. Let $A$ be an $n\times n$ random matrix with i.i.d real subgaussian entries of zero mean and unit variance. We show that with probability at least $1-e^{-\log^{2} n}$ $$ \min\limits_{I\subset[n],\,|I|= m}\|{\bf v}_I\| \geq \frac{m^{3/2}}{n^{3/2}\log^Cn}\|{\bf v}\| $$ for any real eigenvector ${\bf v}$ and any $m\in[\log^C n,n]$, where ${\bf v}_I$ denotes the restriction of ${\bf v}$ to $I$. Further, when the entries of $A$ are complex, with i.i.d real and imaginary parts, we show that with probability at least $1-e^{-\log^{2} n}$ all eigenvectors of $A$ are delocalized in the sense that $$ \min\limits_{I\subset[n],\,|I|= m}\|{\bf v}_I\| \geq \frac{m}{n\log^Cn}\|{\bf v}\| $$ for all $m\in[\log^C{n},n]$. Comparing with related results, in the range $m\in[\log^{C'}{n},n/\log^{C'}{n}]$ in the i.i.d setting and with weaker probability estimates, our lower bounds on $\|{\bf v}_I\|$ strengthen an earlier estimate $\min\limits_{|I|= m}\|{\bf v}_I\| \geq c(m/n)^6\|{\bf v}\| $ obtained in [M. Rudelson, R. Vershynin, Geom. Func. Anal., 2016], and bounds $\min\limits_{|I|= m}\|{\bf v}_I\| \geq c(m/n)^2\|{\bf v}\|$ (in the real setting) and $\min\limits_{|I|= m}\|{\bf v}_I\| \geq c(m/n)^{3/2}\|{\bf v}\|$ (in the complex setting) established in [K. Luh, S. O'Rourke, arXiv:1810.00489]. As the case of real and complex Gaussian matrices shows, our bounds are optimal up to the polylogarithmic multiples. We derive stronger estimates without the polylogarithmic error multiples for null vectors of real $(n-1)\times n$ random matrices.