Infinite random matrices & ergodic decomposition of finite and infinite Hua-Pickrell measures

The ergodic decomposition of a family of Hua-Pickrell measures on the space of infinite Hermitian matrices is studied. By combining previous results of Borodin-Olshanski and our new results, we obtain the first complete description of the ergodic decomposition of Hua-Pickrell measures. First, we show that the ergodic components of any Hua-Pickrell probability measure have no Gaussian factors. Secondly, we show that the sequence of asymptotic eigenvalues of Hua-Pickrell random matrices is balanced in a certain sense and has a “principal value” which coincides with the parameter that reflects the presence of Dirac factor in an ergodic component. This allows us to identify the ergodic decomposition of any Hua-Pickrell probability with a certain determinantal point process with hypergeometric kernel as introduced by Borodin-Olshanski. Finally, we extend the aforesaid results to the case of infinite Hua-Pickrell measures. By using the theory of infinite determinantal measures recently introduced by A.I. Bufetov, we are able to identify the ergodic decomposition of Hua-Pickrell infinite measure with a certain infinite determinantal measure. This resolves a problem of Borodin and Olshanski.