Distribution of the Diagonal Entries of the Resolvent of a Complex Ginibre Matrix

Abstract

The study of eigenvalue distributions in random matrix theory is often conducted by analyzing the resolvent matrix ${\mathbf{G}}_{\mathbf{M}}^N(z)=(z1-\mathbf{M}{)}^{-1}$. The normalized trace of the resolvent, known as the Stieltjes transform ${\mathfrak{g}}_{\mathbf{M}}^N(z)$, converges to a limit ${\mathfrak{g}}_{\mathbf{M}}(z)$ as the matrix dimension $N$ grows, which provides the eigenvalue density ${\rho }_{\mathbf{M}}$ in the large-$N$ limit. In the Hermitian case, the distribution of ${\mathfrak{g}}_{\mathbf{M}}^N(z)$, now regarded as a random variable, is explicitly known when $z$ lies within the limiting spectrum, and it coincides with the distribution of any diagonal entry of ${\mathbf{G}}_{\mathbf{M}}^N(z)$. In this paper, we investigate what becomes of these results when $\mathbf{M}$ is non-Hermitian. Our main result is the exact computation of the diagonal elements of ${\mathbf{G}}_{\mathbf{M}}^N(z)$ when $\mathbf{M}$ is a Ginibre matrix of size $N$, as well as the high-dimensional limit for different regimes of $z$, revealing a tail behavior connected to the statistics of the left and right eigenvectors. Interestingly, the limit distribution is stable under inversion, a property previously observed in the symmetric case. We then propose two general conjectures regarding the distribution of the diagonal elements of the resolvent and its normalized trace in the non-Hermitian case, both of which reveal a symmetry under inversion.