Abstract : This talk will be a general presentation on eigenvectors of non-Hermitian models of random matrices, especially the Gaussian ensembles. We will start by reviewing what is known on the complex Ginibre ensemble, and especially the results by Chalker and Mehlig (2000) and Bourgade and Dubach (2018) on the overlaps between left and right eigenvectors. The case of the real Ginibre ensemble is surprisingly different. I will describe the law of the real Schur decomposition of real Ginibre matrices (due to Edelman, 1997) and how it can be used to study certain statistics of the eigenvectors, notably (i) the overlaps between eigenvectors associated to real eigenvalues (recent results due to Würfel, Crumpton and Fyodorov, 23+) and (ii) the inverse kurtosis of every bulk eigenvector, which is a measure their level of delocalization. In particular I will show that, even though all the eigenvectors are delocalized, those associated to eigenvalues close to the real axis are slightly more localized, and I will give the exact limiting distribution of the inverse kurtosis in the thermodynamic limit. This is based on ongoing work with Lucas Benigni and Guillaume Dubach.
Guillaume Barraquand et Raphaël Butez
