# On delocalization of eigenvectors of random non-Hermitian matrices

Research paper by **Anna Lytova, Konstantin Tikhomirov**

Indexed on: **03 Oct '18**Published on: **03 Oct '18**Published in: **arXiv - Mathematics - Probability**

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### 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 $1-n^{-100}$ $$ \min\limits_{I\subset[n],\,|I|= m}\|{\bf
v}_I\|_2\geq \frac{m^{3/2}}{n^{3/2}\log^Cn}\|{\bf v}\|_2 $$ 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 $1-n^{-100}$ all eigenvectors of
$A$ are delocalized in the sense that $$ \min\limits_{I\subset[n],\,|I|=
m}\|{\bf v}_I\|_2\geq \frac{m}{n\log^Cn}\|{\bf v}\|_2 $$ for all
$m\in[\log^C{n},n]$.
As the case of real and complex Gaussian matrices shows, the above estimates
are optimal up to the polylogarithmic multiples. We derive stronger bounds for
null vectors of real rectangular random matrices.