Three-dimensional Gaussian fluctuations of spectra of overlapping stochastic Wishart matrices

TL;DR

This paper extends previous results on eigenvalue fluctuations of overlapping Wishart matrices, demonstrating that under stochastic evolution, these fluctuations converge to a three-dimensional Gaussian field with an explicit integral representation.

Contribution

It introduces a new result showing that stochastic evolution of Wishart matrices leads to three-dimensional Gaussian fluctuations, extending prior two-dimensional results.

Findings

Fluctuations converge to a 3D Gaussian field.

Explicit contour integral formula for the fluctuations.

Analogy with stochastic Wigner matrices.

Abstract

In arXiv:1410.7268v3, the authors consider eigenvalues of overlapping Wishart matrices and prove that its fluctuations asymptotically convergence to the Gaussian free field. In this brief note, their result is extended to show that when the matrix entries undergo stochastic evolution, the fluctuations asymptotically converge to a three-dimensional Gaussian field, which has an explicit contour integral formula. This is analogous to the result of arXiv:1011.3544 for stochastic Wigner matrices.