Optimal decay of eigenvector overlap for non-Hermitian random matrices

Abstract: We consider the standard overlap $\mathcal{O}_{ij}: =\langle \mathbf{r}_j, \mathbf{r}_i\rangle\langle \mathbf{l}_j, \mathbf{l}_i\rangle$ of any bi-orthogonal family of left and right eigenvectors of a large random matrix $X$ with centred i.i.d. entries and we prove that it decays as an inverse second power of the distance between the corresponding eigenvalues. This extends similar results for the complex Gaussian ensemble from Bourgade and Dubach [arXiv:1801.01219], as well as Benaych-Georges and Zeitouni [arXiv:1806.06806], to any i.i.d. matrix ensemble in both symmetry classes. As a main tool, we prove a two-resolvent local law for the Hermitisation of $X$ uniformly in the spectrum with optimal decay rate and optimal dependence on the density near the spectral edge.