Complex symmetric, self-dual, and Ginibre random matrices: Analytical results for three classes of bulk and edge statistics

TL;DR

This paper analytically compares three classes of non-Hermitian Gaussian random matrices, confirming they have distinct local spectral statistics at bulk and edges, supporting a recent universality conjecture.

Contribution

It derives explicit formulas for characteristic polynomial expectations in three matrix classes and analyzes their spectral statistics, revealing differences and similarities.

Findings

Distinct local bulk and edge spectral statistics for the three classes.

Explicit expressions for characteristic polynomial expectations for finite and infinite matrices.

Effective Lagrangians for the associated non-linear sigma models are derived, showing universality in their form.

Abstract

Recently, a conjecture about the local bulk statistics of complex eigenvalues has been made based on numerics. It claims that there are only three universality classes, which have all been observed in open chaotic quantum systems. Motivated by these new insights, we compute and compare the expectation values of $k$ pairs of complex conjugate characteristic polynomials in three ensembles of Gaussian non-Hermitian random matrices representative for the three classes: the well-known complex Ginibre ensemble, complex symmetric and complex self-dual matrices. In the Cartan classification scheme of non-Hermitian random matrices they are labelled as class A, AI$^\dag$ and AII$^\dag$, respectively. Using the technique of Grassmann variables, we derive explicit expressions for a single pair of expected characteristic polynomials for finite as well as infinite matrix dimension. For the latter we consider the global limit as well as zoom into the edge and the bulk of the spectrum, providing new analytical results for classes AI$^\dag$ and AII$^\dag$. For general $k$, we derive the effective Lagrangians corresponding to the non-linear $\sigma$-models in the respective physical systems. Interestingly, they agree for all three ensembles, while the corresponding Goldstone manifolds, over which one has to perform the remaining integrations, are different and equal the three classical compact groups in the bulk. In particular, our analytical results show that these three ensembles have indeed different local bulk and edge spectral statistics, corroborating the conjecture further.