Mean eigenvector self-overlap in elliptic Ginibre ensembles at strong and weak non-Hermiticity
A talk in the Seminar Zufallsmatrizen series by
Mark Crumpton
| Abstract: | The matrix of eigenvector overlaps, introduced by Chalker $\&$ Mehlig, is known to have many applications, including the description of decay laws in quantum chaotic scattering and the characterization of eigenvalue sensitivity. For normal matrices, the corresponding eigenvector overlaps are trivial due to orthogonality. However, when one considers non-normal matrices, the entries of the matrix of overlaps can become macroscopically large. In this talk, we study the diagonal entries of the matrix of overlaps, denoted self overlaps, for ensembles of $N \times N$ real and complex random matrices with varying degrees of non-normality. We focus in particular on the real and complex elliptic Ginibre ensembles, with mean zero i.i.d. Gaussian entries and a correlation between offdiagonal matrix entries, governed by $\tau \in [0, 1)$. We will present new results for the mean self-overlap associated with complex eigenvalues at finite N in both ensembles, however we are mainly concerned with large N asymptotic behaviour. As N becomes large, we consider three different regions of the complex plane with different density of complex eigenvalues: the spectral bulk, the spectral edge and a region of eigenvalue depletion close to the real line. This is done for two different limits of $\tau$, known as strong non-Hermiticity, where $\tau \in [0, 1)$ is fixed as $\tau \rightarrow \infty$ and weak non-Hermiticity, where $\tau \rightarrow 1$ as $N \rightarrow \infty$. As part of this talk we will also review some important existing results in this field and provide numerical evidence of our new results. Within the CRC this talk is associated to the project(s): C6 |