Phase Transition of Eigenvalues in Deformed Ginibre Ensembles II: GinSE

Consider a random matrix of size 2N as an additive deformation of the quaternionic Ginibre ensemble under a deterministic matrix \(X_0\) $X_0$ with a finite rank, independent of N. When some eigenvalues of \(X_0\) $X_0$ lie inside the unit disk, we prove that local eigenvalue statistics at the real edge is governed by a new class of Pfaffian point processes, for which correlation kernels only depend on geometric multiplicity of eigenvalue, while at the complex edge it is governed by the same statistics as recently found in the deformed GinUE.