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

## TL;DR

This paper explores the hierarchical structure of gamma kernels in determinantal point processes, revealing how measures and Palm distributions relate through explicit Radon-Nikodym derivatives and perturbations.

## Contribution

It establishes a hierarchy of gamma kernels, showing their perturbation relations and explicit Radon-Nikodym derivatives for associated measures and Palm distributions.

## Key findings

- Gamma kernels form a hierarchy with explicit perturbation relations.
- One-point Palm distributions are absolutely continuous with respect to neighboring kernels.
- Radon-Nikodym derivatives are given by normalized multiplicative functionals.

## 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.