Universality at weak and strong non-Hermiticity beyond the elliptic Ginibre ensemble

Abstract

We consider non-Gaussian extensions of the elliptic Ginibre ensemble of complex non-Hermitian random matrices by fixing the trace $\mathrm{Tr}(X{X}^{\ast })$ of the matrix $X$ with a hard or soft constraint. These ensembles have correlated matrix entries and non-determinantal joint densities of the complex eigenvalues. We study global and local bulk statistics in these ensembles, in particular in the limit of weak non-Hermiticity introduced by Fyodorov, Khoruzhenko and Sommers. Here, the support of the limiting measure collapses to the real line. This limit was motivated by physics applications and interpolates between the celebrated sine and Ginibre kernel. Our results constitute a first proof of universality of the interpolating kernel. Furthermore, in the limit of strong non-Hermiticity, where the support of the limiting measure remains an ellipse, we obtain local Ginibre statistics in the bulk of the spectrum.

arXiv commentv1: 22 pages; v2: 30 pages, substantially extended to include fixed trace ensembles

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Cited by in corpus (13)

• Non-Hermitian Physics
• A non-Hermitian generalisation of the Marchenko-Pastur distribution: from the circular law to multi-criticality
• The Elliptic Ginibre Ensemble: A Unifying Approach to Local and Global Statistics for Higher Dimensions
• The high temperature crossover for general 2D Coulomb gases
• Two-Dimensional Elliptic Determinantal Point Processes and Related Systems
• Families of two-dimensional Coulomb gases on an ellipse: correlation functions and universality
• Finite size corrections for real eigenvalues of the elliptic Ginibre matrices
• Wronskian structures of planar symplectic ensembles
• Products of random matrices from fixed trace and induced Ginibre ensembles
• Lemniscate ensembles with spectral singularity
• Real eigenvalues of elliptic random matrices
• Dynamical Universality for Random Matrices
• Singular Value Statistics for the Spiked Elliptic Ginibre Ensemble