Three topological phases of the elliptic Ginibre ensembles with a point charge

Abstract

We consider the complex and symplectic elliptic Ginibre matrices of size $(c+1)N \times (c+1)N$, conditioned to have a deterministic eigenvalue at $ p \in \mathbb{R} $ with multiplicity $ c N $. We show that their limiting spectrum is either simply connected, doubly connected, or composed of two disjoint simply connected components. Moreover, denoting by $\tau \in [0,1]$ the non-Hermiticity parameter, we explicitly characterise the regions in the parameter space $ (p, c, \tau) $ where each topological type emerges. For cases where the droplet is either simply or doubly connected, we provide an explicit description of the limiting spectrum and the corresponding electrostatic energies. As an application, we derive the asymptotic behaviour of the moments of the characteristic polynomial for elliptic Ginibre matrices in the exponentially varying regime.