On eigenvalues of the Brownian sheet matrix

TL;DR

This paper investigates the eigenvalues of matrices formed from Brownian sheets, deriving stochastic PDEs, proving tightness and convergence of spectral measures, and establishing PDEs for the limiting spectral distribution.

Contribution

It introduces stochastic PDEs for eigenvalues of Brownian sheet matrices and characterizes the limiting spectral measure as dimension grows.

Findings

Eigenvalues satisfy a system of stochastic PDEs.

Spectral measures are tight and converge to a limit.

Limiting spectral measure satisfies specific PDEs.

Abstract

We derive a system of stochastic partial differential equations satisfied by the eigenvalues of the symmetric matrix whose entries are the Brownian sheets. We prove that the sequence $\left\{L_{d}(s,t), (s,t)\in[0,S]\times [0,T]\right\}_{d\in\mathbb N}$ of empirical spectral measures of the rescaled matrices is tight on $C([0,S]\times [0,T], \mathcal P(\mathbb R))$ and hence is convergent as $d$ goes to infinity by Wigner's semicircle law. We also obtain PDEs which are satisfied by the high-dimensional limiting measure.