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Working paper

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

arxiv.org. math. Cornell University, 2019. No. 1904.13371 .
Olshanski G., Bufetov A.
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 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 …,K(z−1,z′−1),K(z,z′),K(z+1,z′+1),…, 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.