Condition numbers for real eigenvalues of real elliptic ensemble: weak non-normality at the edge

Abstract: Sensitivity of an eigenvalue $\lambda_i$ to the perturbation of matrix elements is controlled by the eigenvalue condition number defined as $\kappa_i = \sqrt{\left< L_i | L_i\right> \left< R_i|R_i \right> }$, where $\left<L_i\right|$ and $\left|R_i\right>$ are left and right eigenvectors to the eigenvalue $\lambda_i$. In Random Matrix Theory the squared eigenvalue condition number is also known as the eigenvector self-overlap. In this work we calculate the asymptotics of the joint probability density function of the real eigenvalue and the square of the corresponding eigenvalue condition number for the real elliptic ensemble in the double scaling regime of almost Hermiticity and close to the edge of the spectrum. As a byproduct, we also calculate the one-parameter deformation of the Scorer's function.