Phase transition of eigenvalues in deformed Ginibre ensembles

TL;DR

This paper studies phase transitions in eigenvalue distributions of deformed Ginibre matrices, revealing new outlier behaviors and a novel class of edge statistics linked to the Jordan form of the deformation.

Contribution

It characterizes the eigenvalue phase transition and introduces a new class of determinantal point processes for the spectral edge in non-Hermitian random matrices.

Findings

Outlier eigenvalues depend on the Jordan form of the deformation.

New determinantal point processes describe eigenvalue statistics at the spectral edge.

Results extend to real quaternion Ginibre ensembles.

Abstract

Consider a random matrix of size $N$ as an additive deformation of the complex Ginibre ensemble under a deterministic matrix $X_0$ with a finite rank, independent of $N$. When some eigenvalues of $X_0$ separate from the unit disk, outlier eigenvalues may appear asymptotically in the same locations, and their fluctuations exhibit surprising phenomena that highly depend on the Jordan canonical form of $X_0$. These findings are largely due to Benaych-Georges and Rochet \cite{BR}, Bordenave and Capitaine \cite{BC16}, and Tao \cite{Ta13}. When all eigenvalues of $X_0$ lie inside the unit disk, we prove that local eigenvalue statistics at the spectral edge form a new class of determinantal point processes, for which correlation kernels are characterized in terms of the repeated erfc integrals. This thus completes a non-Hermitian analogue of the BBP phase transition in Random Matrix Theory. Similar results hold for the deformed real quaternion Ginibre ensemble.