Mean left-right eigenvector self-overlap in the real Ginibre ensemble (Q6651138)

Given a \(N\times N\) (real) complex random matrix \(X\) and assuming that all its \(N\) eigenvalues have multiplicity one, let denote by \(\mathbf{x}_{L_{m}}, \mathbf{x}_{R_{m}}, m=1, \dots, N, \) the corresponding left and right eigenvectors. We can associate the so called overlap matrix \(\mathcal{O}\) of \(X\) whose entries are \(\mathcal{O}_{n,m} = (\mathbf{x}_{L_{n}}^{\dag}\mathbf{x}_{L_{m}}) (\mathbf{x}_{R_{m}}^{\dag}\mathbf{x}_{R_{n}}), n, m=1, \dots, N.\) In the physical literature the diagonal entries,\(\mathcal{O}_{n,n},\) are generally known as diagonal overlaps, or self-overlaps.\N\NIf \(X\) is a random matrix from a specified probability measure, the statistics of the entries of the overlap matrix become an important object of study. The simplest nontrivial choice is to assume that all entries of \(X\) are mean-zero and independent, identically distributed Gaussian numbers, which can be real, complex, or quaternion. Thus you get the three classical Ginibre ensembles, see [\textit{J. Ginibre}, J. Math. Phys. 6, 440--449 (1965; Zbl 0127.39304)]. The statistics of the overlap matrix in this setting has been influenced by the seminal papers of Chalker-Mehlig, in particular see [\textit{B. Mehlig} and \textit{J. T. Chalker}, J. Math. Phys. 41, No. 5, 3233--3256 (2000; Zbl 0977.82023)], where the complex Ginibre ensemble is analyzed taking into account the statistics of the single-point (mean overlap) \(\mathcal{O}(z)\) and two-point correlation functions \(\mathcal{O}(z_{1}, z_{2})\) associated with the eigenvalues of \(X.\)\N\NIn the paper under the review, the authors focus the attention on the Chalker-Mehlig mean overlap associated with a complex eigenvalue of an \(N \times N\) matrix \(X\) in the real Ginibre ensemble. A general finite \(N\) expression for the mean overlap is deduced. Several scaling regimes when \(N\rightarrow \infty\) are studied. Indeed, in the generic spectral bulk and edge of the real Ginibre ensemble, the limiting expressions for \(\mathcal{O}(z)\) are found to coincide with the known results for the complex Ginibre ensemble but in the region of eigenvalue depletion close to the real axis the asymptotic for the real Ginibre ensemble is quite different.\N\NThe distribution of diagonal overlaps is illustrated from a numerical point of view. Finally, it is conjectured that such a distribution is the same in the bulk and at the edge of both the real and complex Ginibre ensembles, but essentially different in the depletion region of the real Ginibre ensembles.