Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble

We study the distribution of the eigenvalue condition numbers $\kappa_i=\sqrt{ (\mathbf{l}_i^* \mathbf{l}_i)(\mathbf{r}_i^* \mathbf{r}_i)}$ associated with real eigenvalues $\lambda_i$ of partially asymmetric $N\times N$ random matrices from the real Elliptic Gaussian ensemble. The large values of $\kappa_i$ signal the non-orthogonality of the (bi-orthogonal) set of left $\mathbf{l}_i$ and right $\mathbf{r}_i$ eigenvectors and enhanced sensitivity of the associated eigenvalues against perturbations of the matrix entries. We derive the general finite $N$ expression for the joint density function(JDF) ${\cal P}_N(z,t)$ of $t=\kappa_i^2-1$ and $\lambda_i$ taking value $z$, and investigate its several scaling regimes in the limit $N\to \infty$. When the degree of asymmetry is fixed as $N\to \infty$, the number of real eigenvalues is $O(\sqrt{N})$, and in the bulk of the real spectrum $t_i=O(N)$, while on approaching the spectral edges the non-orthogonality is weaker: $t_i=O(\sqrt{N})$. In both cases the corresponding JDFs, after appropriate rescaling, coincide with those found in the earlier studied case of fully asymmetric (Ginibre) matrices. A different regime of weak asymmetry arises when a finite fraction of $N$ eigenvalues remain real as $N\to \infty$. In such a regime eigenvectors are weakly non-orthogonal, $t=O(1)$, and we derive the associated JDF, finding that the characteristic tail ${\cal P}(z,t)\sim t^{-2}$ survives for arbitrary weak asymmetry. As such, it is the most robust feature of the condition number density for real eigenvalues of asymmetric matrices.