Distributions on Partitions, Point Processes,¶ and the Hypergeometric Kernel

By Alexei Borodin, Grigori Olshanski. 1999

Abstract:We study a 3-parametric family of stochastic point processes on the one-dimensional lattice originated from a remarkable family of representations of the infinite symmetric group. We prove that the correlation functions of the processes are given by determinantal formulas with a certain kernel. The kernel can be expressed through the Gauss hypergeometric function; we call it the hypergeometric kernel.In a scaling limit our processes approximate the processes describing the decomposition of representations mentioned above into irreducibles. As we showed in previous works, the correlation functions of these limit processes also have determinantal form with so-called Whittaker kernel. We show that the scaling limit of the hypergeometric kernel is the Whittaker kernel.integrable operator as defined by Its, Izergin, Korepin, and Slavnov. We argue that the hypergeometric kernel can be considered as a kernel defining a ‘discrete integrable operator’.We also show that the hypergeometric kernel degenerates for certain values of parameters to the Christoffel–Darboux kernel for Meixner orthogonal polynomials. This fact is parallel to the degeneration of the Whittaker kernel to the Christoffel–Darboux kernel for Laguerre polynomials.

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Citations in the corpus

Determinantal Point Processes for Machine Learning (2012, background): “Many good mathematical surveys are now available [Borodin, 2009, Hough et al., 2006, Shirai and Takahashi, 2003a,b, Lyons, 2003, Soshnikov, 2000, Tao, 2009].”
Determinantal point processes (2018, background): “rminantal processes were ﬁrst singled out in [Mac75] to model fermions in thermal equilibrium, cf. [Ben73], and the term ‘fermion’ point processes was used. The term ‘determinantal’ was introduced in [Bor00a], for the reason that the particles of the process studied there were of two kinds; particle”
An introduction to harmonic analysis on the infinite symmetric group (2003, methodology): “singled out by Macchi [49], [50] under the name of fermion processes. The motivation comes from a connection with the fermionic Fock space. The term “determinantal” was suggested in Borodin–Olshanski [15]. We found it more appropriate, because in our concrete situation, point conﬁgurations may be vi”
Determinantal probability measures (2002, background): “Introduction 3 instances have been studied in Borodin (2000), Borodin and Olshanski (2000, 2001, 2002), Borodin, Okounkov, and Olshanski (2000), Johansson (2001, 2002), Okounkov (2001), and Okounkov and Reshetikhin (2001).”
Harmonic analysis on the infinite symmetric group (2003, background): “hat is, all faces Ω(p,q) are null sets with respect to the spectral type of Tz. 0.4. The case z ∈C\Z. This case is studied in detail in a series of papers by Borodin and Olshanski, see [Ol5], [Bor2], [BO3], the expository papers [BO1], [Ol6], and references therein. When z ∈C\Z, there exists a disti”