The Boundary of the Orbital Beta Process

The unitarily invariant probability measures on infinite Hermitian matrices have been classified by Pickrell, and by Olshanski and Vershik. This classification is equivalent to determining the boundary of a certain inhomogeneous Markov chain with given transition probabilities. This formulation of the problem makes sense for general $\beta$-ensembles when one takes as the transition probabilities the Dixon-Anderson conditional probability distribution. In this paper we determine the boundary of this Markov chain for any $\beta \in (0,\infty]$, also giving in this way a new proof of the classical $\beta=2$ case. Finally, as a by-product of our results we obtain alternative proofs of the almost sure convergence of the rescaled Hua-Pickrell and Laguerre $\beta$-ensembles to the general $\beta$ Hua-Pickrell and $\beta$ Bessel point processes respectively.