A hierarchy of Palm measures for determinantal point processes with gamma kernels

Abstract : The gamma kernels are a family of projection kernels $K^{(z,z')}=K^{(z,z')}(x,y)$ on a doubly infinite $1$-dimensional lattice. They are expressed through Euler's gamma function and depend on two continuous parameters $z,z'$. The gamma kernels initially arose from a model of random partitions via a limit transition. On the other hand, these kernels are closely related to unitarizable representations of the Lie algebra $\mathfrak{su}(1,1)$. Every gamma kernel $K^{(z,z')}$ serves as a correlation kernel for a determinantal measure $M^{(z,z')}$, which lives on the space of infinite point configurations on the lattice. We examine chains of kernels of the form $$ \ldots, K^{(z-1,z'-1)}, \; K^{(z,z')},\; K^{(z+1,z'+1)}, \ldots, $$ and establish the following hierarchical relations inside any such chain: Given $(z,z')$, the kernel $K^{(z,z')}$ is a one-dimensional perturbation of (a twisting of) the kernel $K^{(z+1,z'+1)}$, and the one-point Palm distributions for the measure $M^{(z,z')}$ are absolutely continuous with respect to $M^{(z+1,z'+1)}$. We also explicitly compute the corresponding Radon-Nikod\'ym derivatives and show that they are given by certain normalized multiplicative functionals.
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https://hal.archives-ouvertes.fr/hal-02438985
Contributor : Sergey Berezin <>
Submitted on : Tuesday, January 14, 2020 - 2:12:00 PM
Last modification on : Thursday, January 23, 2020 - 6:22:13 PM

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  • HAL Id : hal-02438985, version 1
  • ARXIV : 1904.13371

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Alexander I. Bufetov, Grigori Olshanski. A hierarchy of Palm measures for determinantal point processes with gamma kernels. 2020. ⟨hal-02438985⟩

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