$J$-Hermitian determinantal point processes: balanced rigidity and balanced Palm equivalence

TL;DR

This paper investigates special determinantal point processes with $J$-Hermitian kernels, establishing criteria for balanced rigidity and Palm equivalence, and applies these results to processes with Whittaker kernels.

Contribution

It formulates general conditions for $J$-Hermitian determinantal processes to exhibit balanced rigidity and Palm equivalence, and demonstrates these properties for processes with Whittaker kernels.

Findings

Balanced rigidity is established for certain determinantal processes.

Balanced Palm equivalence property is proven for processes with Whittaker kernels.

General criteria for these properties are formulated and verified.

Abstract

We study Palm measures of determinantal point processes with $J$-Hermitian correlation kernels. A point process $\mathbb{P}$ on the punctured real line $\mathbb{R}^* = \mathbb{R}_{+} \sqcup \mathbb{R}_{-}$ is said to be $\textit{balanced rigid}$ if for any precompact subset $B\subset \mathbb{R}^*$, the $\textit{difference}$ between the numbers of particles of a configuration inside $B\cap \mathbb{R}_{+}$ and $B\cap \mathbb{R}_{-}$ is almost surely determined by the configuration outside $B$. The point process $\mathbb{P}$ is said to have the $\textit{balanced Palm equivalence property}$ if any reduced Palm measure conditioned at $2n$ distinct points, $n$ in $\mathbb{R}_{+}$ and $n$ in $\mathbb{R}_{-}$, is equivalent to the $\mathbb{P}$. We formulate general criteria for determinantal point processes with $J$-Hermitian correlation kernels to be balanced rigid and to have the balanced Palm equivalence property and prove, in particular, that the determinantal point processes with Whittaker kernels of Borodin and Olshanski are balanced rigid and have the balanced Palm equivalence property.