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

Abstract

The gamma kernels are a family of projection kernels $K^{(z,z^{\prime })}=K^{(z,z^{\prime })}(x,y)$ on a doubly infinite $1$-dimensional lattice. They are expressed through Euler's gamma function and depend on two continuous parameters $z,z^{\prime }$. 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 $K^{(z,z^{\prime })}$ serves as a correlation kernel for a determinantal measure $M^{(z,z^{\prime })}$, which lives on the space of infinite point configurations on the lattice. We examine chains of kernels of the form $$\dots ,K^{(z-1,z^{\prime }-1)},K^{(z,z^{\prime })},K^{(z+1,z^{\prime }+1)},\dots ,$$ and establish the following hierarchical relations inside any such chain: Given $(z,z^{\prime })$, the kernel $K^{(z,z^{\prime })}$ is a one-dimensional perturbation of (a twisting of) the kernel $K^{(z+1,z^{\prime }+1)}$, and the one-point Palm distributions for the measure $M^{(z,z^{\prime })}$ are absolutely continuous with respect to $M^{(z+1,z^{\prime }+1)}$. We also explicitly compute the corresponding Radon-Nikodým derivatives and show that they are given by certain normalized multiplicative functionals.

Version 2: minor changes, typos fixed