Determinantal point processes with J-Hermitian correlation kernels

Let X be a locally compact Polish space and let m be a reference Radon measure on X. Let ${\mathrm{\Gamma }}_X$ 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 ${\mathrm{\Gamma }}_X$. A point process $\mu$ is called determinantal if its correlation functions have the form $k^{(n)}(x_1,\dots ,x_n)=\det [K(x_i,x_j){]}_{i,j=1,\dots ,n}$. 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: $X=X_1\bigsqcup X_2$. 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).