# Moscow Mathematical Journal

Volume 21, Issue 4, October–December 2021 pp. 659–694.

The Boundary of the Orbital Beta Process

**Authors**:
Theodoros Assiotis (1) and Joseph Najnudel (2)

**Author institution:**(1) School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Rd, Edinburgh EH9 3FD, U.K.

(2) Laboratoire Mathématiques & Interactions J.A. Dieudonné – Université Côte d'Azur – CNRS UMR 7351 – Parc Valrose 06108 NICE CEDEX 2, France

**Summary: **

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 (Pickrell, Olshanski and Vershik). 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; these results were obtained earlier by Killip and Stoiciu, Valkó and Virág, Ramírez and Rider.

2020 Math. Subj. Class. 60B20, 60F15, 60J05, 60J50.

**Keywords:**Infinite random matrices, beta ensembles, ergodic measures, boundary of Markov chains.

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