Singular value statistics for the spiked elliptic Ginibre ensemble

The complex elliptic Ginibre ensemble with coupling constant τ is a complex Gaussian matrix interpolating between the Gaussian Unitary Ensemble (GUE) and the complex Ginibre ensemble. It has been known for some time that its eigenvalues form a determinantal point process in the complex plane. Until recently, however, Kanazawa and Kieburg [35] have proved that the singular values form a Pfaffian point process. In this paper we consider an extended complex elliptic Ginibre ensemble, which connects the GUE and the spiked complex Wishart ensemble, and investigate local statistics of the singular values. Particularly as τ 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 FGUE and FGUE2 where FGUE is the GUE Tracy-Widom distribution.