Parameter symmetry in perturbed GUE corners process and reflected drifted Brownian motions

TL;DR

This paper explores the symmetry properties of eigenvalues in perturbed GUE matrices and their connection to reflected Brownian motions with specific drifts, introducing new Markov transition techniques.

Contribution

It introduces Markov transitions based on exponential eigenvalue jumps and establishes a distributional symmetry linking matrix perturbations to reflected Brownian motions.

Findings

Eigenvalue distributions are invariant under certain Markov transitions.

A new symmetry for reflected Brownian motions with arithmetic progression drifts.

Connection between matrix perturbations and stochastic process symmetries.

Abstract

The perturbed GUE corners ensemble is the joint distribution of eigenvalues of all principal submatrices of a matrix $G+\mathrm{diag}(\mathbf{a})$, where $G$ is the random matrix from the Gaussian Unitary Ensemble (GUE), and $\mathrm{diag}(\mathbf{a})$ is a fixed diagonal matrix. We introduce Markov transitions based on exponential jumps of eigenvalues, and show that their successive application is equivalent in distribution to a deterministic shift of the matrix. This result also leads to a new distributional symmetry for a family of reflected Brownian motions with drifts coming from an arithmetic progression. The construction we present may be viewed as a random matrix analogue of the recent results of the first author and Axel Saenz (arXiv:1907.09155 [math.PR]).