Universal shocks in the Wishart random-matrix ensemble - a sequel

TL;DR

This paper investigates the behavior of eigenvalues in large complex Wishart matrices, deriving PDEs and discovering universal shock phenomena near spectral edges, including a critical point at the origin characterized by a Bessoid function.

Contribution

It introduces a PDE framework for the averaged characteristic polynomial of Wishart matrices and uncovers universal shock behavior and a Bessoid function near a critical spectral point.

Findings

Derivation of a PDE for the averaged characteristic polynomial.

Identification of shock phenomena at spectral edges.

Discovery of a Bessoid function describing universal behavior near the critical point.

Abstract

We study the diffusion of complex Wishart matrices and derive a partial differential equation governing the behavior of the associated averaged characteristic polynomial. In the limit of large size matrices, the inverse Cole-Hopf transform of this polynomial obeys a nonlinear partial differential equation whose solutions exhibit shocks at the evolving edges of the eigenvalue spectrum. In a particular scenario one of those shocks hits the origin that plays the role of an impassable wall. To investigate the universal behavior in the vicinity of this wall, a critical point, we derive an integral representation for the averaged characteristic polynomial and study its asymptotic behavior. The result is a Bessoid function.