Non-Hermitean Wishart random matrices (I)

TL;DR

This paper introduces a non-Hermitean extension of Wishart random matrices to analyze complex stochastic time series from remote systems, deriving spectral properties and an analogue of the Marchenko-Pastur law.

Contribution

It develops a spectral theory for non-Hermitean Wishart matrices and establishes an asymptotic eigenvalue density, connecting to models in quantum chromodynamics.

Findings

Derived a complex-plane analogue of the Marchenko-Pastur law.

Provided a detailed spectral analysis of non-Hermitean Wishart matrices.

Identified a connection with matrix models in quantum chromodynamics.

Abstract

A non-Hermitean extension of paradigmatic Wishart random matrices is introduced to set up a theoretical framework for statistical analysis of (real, complex and real quaternion) stochastic time series representing two "remote" complex systems. The first paper in a series provides a detailed spectral theory of non-Hermitean Wishart random matrices composed of complex valued entries. The great emphasis is placed on an asymptotic analysis of the mean eigenvalue density for which we derive, among other results, a complex-plane analogue of the Marchenko-Pastur law. A surprising connection with a class of matrix models previously invented in the context of quantum chromodynamics is pointed out.