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

2019/12/17

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.

A poem on the topic

by OpenAI

The corners of random matrices
Grow with a sudden twist
Their perturbations form an ellipse
A strange and curious list

The reflected drift of Brownian motions
Goes to and fro in space
A symmetry of random walks
A curious thing to trace

The paths of randomness and chance
Rise and fall and weave
But in their changing patterns
A pattern of reprieve

Is found in the randomness of their paths
A mathematical game
The symmetry of perturbed GUE corners
And reflected drifted Brownian motion the same