The z-measures on partitions, Pfaffian point processes, and the matrix hypergeometric kernel

Eugene Strahov

doi:10.1016/j.aim.2009.11.008

The z-measures on partitions, Pfaffian point processes, and the matrix hypergeometric kernel

Eugene Strahov^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

We consider a point process on one-dimensional lattice originated from the harmonic analysis on the infinite symmetric group, and defined by the z-measures with the deformation (Jack) parameter 2. We derive an exact Pfaffian formula for the correlation function of this process. Namely, we prove that the correlation function is given as a Pfaffian with a 2 × 2 matrix kernel. The kernel is given in terms of the Gauss hypergeometric functions, and can be considered as a matrix analogue of the Hypergeometric kernel introduced by A. Borodin and G. Olshanski (2000) [5]. Our result holds for all values of admissible complex parameters.

Original language	English
Pages (from-to)	130-168
Number of pages	39
Journal	Advances in Mathematics
Volume	224
Issue number	1
DOIs	https://doi.org/10.1016/j.aim.2009.11.008
State	Published - 1 May 2010

Bibliographical note

Funding Information:
✩ Supported by US–Israel Binational Science Foundation (BSF), Grant No. 2006333, and by Israel Science Foundation (ISF), Grant No. 0397937. E-mail address: [email protected].

Keywords

Correlation functions
Pfaffian point processes
Random partitions
The Meixner orthogonal polynomials
Young diagrams

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10.1016/j.aim.2009.11.008

Cite this

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abstract = "We consider a point process on one-dimensional lattice originated from the harmonic analysis on the infinite symmetric group, and defined by the z-measures with the deformation (Jack) parameter 2. We derive an exact Pfaffian formula for the correlation function of this process. Namely, we prove that the correlation function is given as a Pfaffian with a 2 × 2 matrix kernel. The kernel is given in terms of the Gauss hypergeometric functions, and can be considered as a matrix analogue of the Hypergeometric kernel introduced by A. Borodin and G. Olshanski (2000) [5]. Our result holds for all values of admissible complex parameters.",

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N2 - We consider a point process on one-dimensional lattice originated from the harmonic analysis on the infinite symmetric group, and defined by the z-measures with the deformation (Jack) parameter 2. We derive an exact Pfaffian formula for the correlation function of this process. Namely, we prove that the correlation function is given as a Pfaffian with a 2 × 2 matrix kernel. The kernel is given in terms of the Gauss hypergeometric functions, and can be considered as a matrix analogue of the Hypergeometric kernel introduced by A. Borodin and G. Olshanski (2000) [5]. Our result holds for all values of admissible complex parameters.

AB - We consider a point process on one-dimensional lattice originated from the harmonic analysis on the infinite symmetric group, and defined by the z-measures with the deformation (Jack) parameter 2. We derive an exact Pfaffian formula for the correlation function of this process. Namely, we prove that the correlation function is given as a Pfaffian with a 2 × 2 matrix kernel. The kernel is given in terms of the Gauss hypergeometric functions, and can be considered as a matrix analogue of the Hypergeometric kernel introduced by A. Borodin and G. Olshanski (2000) [5]. Our result holds for all values of admissible complex parameters.

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KW - The Meixner orthogonal polynomials

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The z-measures on partitions, Pfaffian point processes, and the matrix hypergeometric kernel

Abstract

Bibliographical note

Keywords

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