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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


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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