The Borodin-Olshanski Problem and Determinantal Point Processes.

Sun, 10/05/2015 - 12:00

Let mu_{m,n} be the canonical invariant measure on the Grassmann manifold
of m-dimensional subspaces in C^{m+n}; the flat coordinates on the Grassmann
manifold allow us to consider mu_{m,n} as a measure on the space Mat(m x n) of
complex matrices. By definition, the family of measures mu_{m,n}  has
the property of consistency under natural projections
Mat((m + 1)  n) ---> Mat(m  n) ; Mat(m x (n + 1)) ---> Mat(m x n)
and consequently defines a probability measure  on the space Mat of infinite
complex matrices. The measure mu is by definition unitarily-invariant and admits
a natural one-parameter family of unitarily-invariant deformations mu^(s), called
the Pickrell measures. The Pickrell measures are finite for s > -1 and infinite
for s < 0.

The first main result of the talk is the solution to the problem, posed by
Borodin and Olshanski in 2000, of the explicit description of the ergodic decomposition of infinite Pickrell measures. The decomposing measures are naturally identified with sigma-finite processes on the half-line R+ and can be viewed as sigma-finite analogues of determinantal point processes. For different values of the parameter s, these measures are mutually singular.

In the second part of the talk we will discuss absolute continuity and singularity of determinantal point processes. The main result here is that determinantal point processes on Z induced by integrable kernels are indeed quasi-invariant under the action of the in nite symmetric group. The Radon-Nikodym derivative is found explicitly. A key example is the discrete sine-process of Borodin, Okounkov and Olshanski. This result has a continuous counterpart: namely, that
determinantal point processes with integrable kernels on R, a class that includes processes arising in random matrix theory such as the sine-process, the process with the Bessel kernel or the Airy kernel, are quasi-invariant under the action of the group of di eomorphisms with compact support.

The first part of the talk is based on the preprint;
the second part, on the preprint