Determinantal point processes with $J$-Hermitian correlation kernels

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 $K(x,y)$.