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

Alexander I. Bufetov, Grigori Olshanski Studia Mathematica MSC: Primary 60G55; Secondary 05E10. DOI: 10.4064/sm210823-10-3 Published online: 28 June 2022

Abstract

The gamma kernels are a family of projection kernels $K^{(z,z’)}= K^{(z,z’)}(x,y)$ on a doubly infinite one-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 $$ \ldots , K^{(z-1,z’-1)}, K^{(z,z’)}, K^{(z+1,z’+1)}, \dots , $$ 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–Nikodým derivatives and show that they are given by certain normalized multiplicative functionals.

Authors

  • Alexander I. BufetovAix-Marseille Université
    Centrale Marseille, CNRS
    Institut de Mathématiques de Marseille, UMR 7373
    39 Rue F. Joliot Curie
    13453 Marseille, France
    and
    Steklov Mathematical Institute of RAS
    Gubkina 8
    Moscow 119991, Russia
    and
    Institute for Information Transmission Problems
    Bolshoy Karetny 19
    Moscow 127051, Russia
    e-mail
  • Grigori OlshanskiInstitute for Information Transmission Problems
    Bolshoy Karetny 19
    Moscow 127051, Russia
    and
    Skolkovo Institute of Science and Technology
    Bolshoy Boulevard 30, bld. 1
    Moscow 121205, Russia
    and
    Faculty of Mathematics, HSE University
    Usacheva 6
    Moscow 119048, Russia
    e-mail

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