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Steklov Mathematical Institute Seminar
November 21, 2013 16:00, Moscow, Steklov Mathematical Institute of RAS, Conference Hall (8 Gubkina)
 


The infinite Bessel point process and unitarily-invariant measures on spaces of infinite matrices

A. I. Bufetov

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Abstract: We start the Jacobi orthogonal polynomial ensemble. It is well-known since the work of Tracy and Widom in 1993 that the Bessel kernel is the scaling limit of the Christoffel-Darboux kernels of the Jacobi polynomials and, consequently, the Jacobi ensemble converge to the Bessel determinantal point process in the space of configurations on the half-line.
We next consider the infinite Jacobi ensemble. The talk will provide an explicit description of its scaling limit — an infinite measure on thespace of configurations called the infinite Bessel point process.
The Bessel kernel induces the operator of orthogonal projection onto the space of functions on the half-line with Hankel transform supported on the unit interval. The infinite Bessel point process is given by a space of locally square-integrable functions obtained as a finite-rank perturbation of the range of the Bessel projector.
Inducing the infinite Bessel point process to the subset of configurations that stay away from zero, one obtains a determinantal probability measure given by a projection operator whose range is found explicitly.
Furthermore, the infinite Bessel process can be represented as the product of a multiplicative functional and a determinantal probability measure corresponding to a projection operator. The ranges of the resulting projection operators are found explicitly, but an explicit formula for the kernels can only be given in a few particular cases.
In the second part of the talk we will see that the infinite Bessel point process naturally appears in the problem, posed by Borodin and Olshanski in 2000, of ergodic decomposition of infinite unitarily-invariant measures on spaces of infinite complex matrices.

References
  1. A. I. Bufetov, “O multiplikativnykh funktsionalakh determinantnykh protsessov”, UMN, 67:1(403) (2012), 177–178  mathnet  crossref  mathscinet  zmath; A. I. Bufetov, “Multiplicative functionals of determinantal processes”, Russian Math. Surveys, 67:1 (2012), 181–182  crossref  mathscinet  zmath  adsnasa  isi  scopus
  2. A. I. Bufetov, “Infinite determinantal measures”, Electron. Res. Announc. Math. Sci., 20 (2013), 12–30  mathscinet  zmath  isi  scopus
  3. A. I. Bufetov, “Finiteness of egodic unitarily invariant measures on spaces of infinite matrices”, Ann. Inst. Fourier (Grenoble), 64 (2014) (to appear)  crossref  zmath; (2011), arXiv: 1108.2737 [math.DS]


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