# Hua-Pickrell diffusions and Feller processes on the boundary of the graph of spectra

## TL;DR

This paper constructs and analyzes diffusion processes on the boundary of a spectral graph, linking them to random matrix spectra and invariant measures, with explicit dynamics for Dyson's Brownian motion.

## Contribution

It introduces new Feller-Markov processes on the spectral boundary, extending Hua-Pickrell measures to infinite dimensions and connecting them to random matrix theory and Gelfand-Tsetlin patterns.

## Key findings

- Processes relate to the Sine_2 point process in the bulk of large random matrices.
- Explicit deterministic systems describe Dyson's Brownian motion on the boundary.
- Invariant measures include Hua-Pickrell and Circular Unitary Ensemble.

## Abstract

We consider consistent diffusion dynamics, leaving the celebrated Hua-Pickrell measures, depending on a complex parameter $s$, invariant. These, give rise to Feller-Markov processes on the infinite dimensional boundary $\Omega$ of the "graph of spectra", the continuum analogue of the Gelfand-Tsetlin graph, via the method of intertwiners of Borodin and Olshanski. In the particular case of $s=0$, this stochastic process is closely related to the $\mathsf{Sine_2}$ point process on $\mathbb{R}$ that describes the spectrum in the bulk of large random matrices. Equivalently, these coherent dynamics are associated to interlacing diffusions in Gelfand-Tsetlin patterns having certain Gibbs invariant measures. Moreover, under an application of the Cayley transform when $s=0$ we obtain processes on the circle leaving invariant the multilevel Circular Unitary Ensemble. We finally prove that the Feller processes on $\Omega$ corresponding to Dyson's Brownian motion and its stationary analogue are given by explicit and very simple deterministic dynamical systems.