Determinantal point processes with J-Hermitian correlation kernels

TL;DR

This paper characterizes the conditions under which determinantal point processes with J-Hermitian correlation kernels exist on split spaces, expanding the understanding of structured point processes in mathematical physics.

Contribution

It provides a necessary and sufficient condition for the existence of determinantal point processes with J-Hermitian kernels, a class not previously fully characterized.

Findings

Derived a criterion for existence of such processes.

Extended the theory of determinantal point processes to J-Hermitian kernels.

Clarified the structure of correlation kernels on split spaces.

Abstract

Let X be a locally compact Polish space and let m be a reference Radon measure on X. Let $\Gamma_X$ denote the configuration space over X, that is, the space of all locally finite subsets of X. A point process on X is a probability measure on $\Gamma_X$. A point process $\mu$ is called determinantal if its correlation functions have the form $k^{(n)}(x_1,\ldots,x_n)=\det[K(x_i,x_j)]_{i,j=1,\ldots,n}$. The function K(x,y) is called the correlation kernel of the determinantal point process $\mu$. Assume that the space X is split into two parts: $X=X_1\sqcup X_2$. A kernel K(x,y) is called J-Hermitian if it is Hermitian on $X_1\times X_1$ and $X_2\times X_2$, and $K(x,y)=-\overline{K(y,x)}$ for $x\in X_1$ and $y\in X_2$. We derive a necessary and sufficient condition of existence of a determinantal point process with a J-Hermitian correlation kernel K(x,y).