Universal shocks in the Wishart random matrix ensemble - 1

TL;DR

This paper derives an exact PDE for the average characteristic polynomial of a diffusing Wishart matrix, revealing shock formation and universal Bessel oscillations in the large N limit.

Contribution

It introduces a new PDE approach for Wishart matrices, generalizing Burgers equation and analyzing finite-size corrections and universal edge phenomena.

Findings

Exact PDE for Wishart characteristic polynomial derivative

Identification of shock singularities and their fluid dynamics analogy

Recovery of universal Bessel oscillations at the spectral edge

Abstract

We show that the derivative of the logarithm of the average characteristic polynomial of a diffusing Wishart matrix obeys an exact partial differential equation valid for an arbitrary value of N, the size of the matrix. In the large N limit, this equation generalizes the simple Burgers equation that has been obtained earlier for Hermitian or unitary matrices. The solution through the method of characteristics presents singularities that we relate to the precursors of shock formation in fluid dynamical equations. The 1/N corrections may be viewed as viscous corrections, with the role of the viscosity being played by the inverse of the doubled dimension of the matrix. These corrections are studied through a scaling analysis in the vicinity of the shocks, and one recovers in a simple way the universal Bessel oscillations (so-called hard edge singularities) familiar in random matrix theory.