Eigenvectors of non normal random matrices

We study the angles between the eigenvectors of a random $n\times n$ complex matrix $M$ with density $\propto \mathrm{e} ^{-n\operatorname{Tr} V(M^*M)}$ and $x\mapsto V(x^2)$ convex. We prove that for unit eigenvectors $\mathbf{v} ,\mathbf{v} '$ associated with distinct eigenvalues $\lambda ,\lambda '$ that are the closest to specified points $z,z'$ in the complex plane, the rescaled inner product \[ \sqrt{n} (\lambda '-\lambda )\langle \mathbf{v} ,\mathbf{v} '\rangle \] is uniformly sub-Gaussian, and give a more precise statement in the case of the Ginibre ensemble.