Infinite determinantal measures and the Borodin–Olshanski problem

Consider the action of the Cartesian square of the infinite unitary group on the space of infinite complex matrices. D. Pickrell in 1990 constructed for this action a one-parameter family of invariant probability measures, a natural analogue of the Haar measure. In 2000 Borodin and Olshanski gave an explicit description of the ergodic decomposition of finite Pickrell measures in terms of determinantal point processes with the Bessel kernel of Tracy and Widom, which is well known in random matrix theory And is the scaling limit of Christoffel–Darboux kernels corresponding to classical Jacobi orthogonal polynomials. Borodin and Olshanski extended Pickrell’s construction, exhibited a family of infinite unitarily invariant measures on the space of infinite complex matrices and posed the question of the description of their ergodic decomposition. The answer will be given in the talk. The main role is played by a construction of sigma-finite analogues of determinantal point processes That are scaling limits of scaling limits of infinite analogues of classical orthogonal polynomial ensembles. The talk is based on the paper [1].

References

1. A. I. Bufetov, “Infinite determinantal measures”, Elec. Res. Ann. Math. Sci., 20 (2013), 12–30 http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=8282, arXiv: 1207.6793