Mark Crumpton

Eigenvector Self-Overlaps In The Ginibre Ensembles And Beyond

The statistics of eigenvectors of non-Hermitian random matrices has attracted growing attention recently due to applications in quantum chaotic scattering and characterising eigenvalue sensitivity. This area of study was introduced in the seminal 1998 work of Chalker & Mehlig, who computed the large matrix size, N, limit of the self-overlap between left and right eigenvectors in Ginibre’s complex ensemble. Since then, these results have been built upon to include higher order statistics and some universal results for the self-overlap in a variety of non-Hermitian random matrix ensembles. In this talk, we aim to review some of these existing results for finite N and asymptotically for large N. The analysis at large N mostly
focuses on elliptic Ginibre matrices, which have mean zero i.i.d. Gaussian entries and a correlation between off-diagonal matrix entries, governed by τ ∈ [0,1). This leads to two different asymptotic limits namely strong non-Hermiticity, where τ ∈ [0,1) is fixed as N → ∞ and weak non-Hermiticity, where τ → 1 as N → ∞. When considering real and complex matri-
ces it is generally found that results at strong non-Hermiticity are universal across the two classes, but the same phenomenon does not hold at weak non-Hermiticity. Finally, we will consider some open problems and potentially interesting areas of future study in the realm of eigenvector self-overlaps.