Optimal Delocalization for Non--Hermitian Eigenvectors

We prove an optimal order delocalization estimate for the eigenvectors of general $N\times N$ non-Hermitian matrices $X$: $\mathrm{\parallel }\mathbf{v}{\mathrm{\parallel }}_{\mathrm{\infty }}\le C\sqrt{\frac{\log N}{N}}$ with very high probability, for any right or left eigenvector $\mathbf{v}$ of $X$. This improves upon the previous tightest bound of Rudelson and Vershynin [arXiv:1306.2887] of $\mathcal{O}((\log N{)}^{9\mathrm{/}2}{N}^{-1\mathrm{/}2})$, and holds under weaker assumptions on the tail of the matrix elements. In addition to the coordinate basis, our bound holds for the ${\mathrm{\ell }}^{\mathrm{\infty }}$ norm in any deterministic orthonormal basis. Our result is proven via a dynamical method, by studying the flow of the resolvent of the Hermitization of $X$ and proving local laws on short scales.