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

Abstract

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.