# The Borodin-Olshanski Problem and Determinantal Point Processes.

Speaker
Alexander I. Bufetov (CNRS, Steklov, IITP, NRU-HSE)
Date
10/05/2015 - 13:00 - 12:00Add to Calendar 2015-05-10 12:00:00 2015-05-10 13:00:00 The Borodin-Olshanski Problem and Determinantal Point Processes. 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 http://arxiv.org/abs/1312.3161; the second part, on the preprint http://arxiv.org/abs/1409.2068. seminar room אוניברסיטת בר-אילן - המחלקה למתמטיקה mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
seminar room
Abstract

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 http://arxiv.org/abs/1312.3161;
the second part, on the preprint http://arxiv.org/abs/1409.2068.

תאריך עדכון אחרון : 05/05/2015