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arXiv · 1905.08684

The boundary of the orbital beta process

Theodoros Assiotis·Joseph Najnudel
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Abstract

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 $β$-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 $β\in (0,\infty]$, also giving in this way a new proof of the classical $β=2$ 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 $β$-ensembles to the general $β$ Hua-Pickrell and $β$ Bessel point processes respectively.

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Theodoros Assiotis, Joseph Najnudel. 2020-08-20. The boundary of the orbital beta process. https://arxiv.org/abs/1905.08684

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