Hydrodynamical spectral evolution for random matrices

TL;DR

This paper develops a hydrodynamical framework to analyze the spectral evolution of random matrices, deriving PDEs that describe eigenvalue densities and connecting them to matrix integral asymptotics.

Contribution

It introduces a macroscopic hydrodynamical approach to derive PDEs for spectral evolution in various Dyson Brownian motion models, including chiral and Wishart matrices.

Findings

Derivation of PDEs from hydrodynamical principles for different Dyson models.

Application of method of characteristics to solve spectral PDEs.

Connection of spectral PDEs to matrix integral asymptotics.

Abstract

The eigenvalues of the matrix structure $X + X^{(0)}$, where $X$ is a random Gaussian Hermitian matrix and $X^{(0)}$ is non-random or random independent of $X$, are closely related to Dyson Brownian motion. Previous works have shown how an infinite hierarchy of equations satisfied by the dynamical correlations become triangular in the infinite density limit, and give rise to the complex Burgers equation for the Green's function of the corresponding one-point density function. We show how this and analogous partial differential equations, for chiral, circular and Jacobi versions of Dyson Brownian motion follow from a macroscopic hydrodynamical description involving the current density and continuity equation. The method of characteristics gives a systematic approach to solving the PDEs, and in the chiral case we show how this efficiently reclaims the characterisation of the global eigenvalue density for non-central Wishart matrices due to Dozier and Silverstein. Collective variables provide another approach to deriving the complex Burgers equation in the Gaussian case, and we show that this approach applies equally as well to chiral matrices. We relate both the Gaussian and chiral cases to the asymptotics of matrix integrals.