Optimal Delocalization for Non--Hermitian Eigenvectors

We prove an optimal order delocalization estimate for the eigenvectors of general $N \times N$ non-Hermitian matrices $X$: $\| {\bf v } \|_\infty \leq C \sqrt{\frac{\log N}{N}}$ with very high probability, for any right or left eigenvector ${\bf v}$ of $X$. This improves upon the previous tightest bound of Rudelson and Vershynin [arXiv:1306.2887] of $\mathcal{O}( ( \log N)^{9/2}N^{-1/2})$, and holds under weaker assumptions on the tail of the matrix elements. In addition to the coordinate basis, our bound holds for the $\ell^\infty$ 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.