A hierarchy of Palm measures for determinantal point processes with gamma kernels
Volume 267 / 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.
