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 Ω 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 Sine2 point process on 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 Ω corresponding to Dyson's Brownian motion and its stationary analogue are given by explicit and very simple deterministic dynamical systems.
|Journal||Annales de l'Institut Henri Poincaré, Probabilités et Statistiques|
|Early online date||16 Mar 2020|
|Publication status||Published - 31 May 2020|