On delocalization of eigenvectors of random non-Hermitian matrices

Research paper by Anna Lytova, Konstantin Tikhomirov

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


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.