Singular Value Statistics for the Spiked Elliptic Ginibre Ensemble

Abstract: The complex elliptic Ginibre ensemble with coupling $\tau$ is a complex Gaussian matrix interpolating between the Gaussian Unitary Ensemble (GUE) and the Ginibre ensemble. It has been known for some time that its eigenvalues form a determinantal point process in the complex plane. A recent result of Kanazawa and Kieburg (arXiv:1804.03985) shows that the singular values form a Pfaffian point process. In this paper we turn to consider an extended elliptic Ginibre ensemble, which connects the GUE and the spiked Wishart matrix, and prove that the singular values still build a Pfaffian point process with correlation kernels expressed by contour integral representations. As $\tau$ tends to 1 at a certain critical rate, we prove that the limiting distribution of the largest singular value is described as a new Fredholm Pfaffian series, which connects two distributions $F_{\mathrm{GUE}}$ and $F^{2}_{\mathrm{GUE}}$ where $F_{\mathrm{GUE}}$ is the GUE Tracy-Widom distribution. For fixed $\tau$, we prove the Baik-Ben Arous-P\'ech\'e transition of the largest singular value and the sine kernel in the bulk. We also observe a crossover phenomenon at the origin when $\tau$ tends to 1 at another critical rate.