Determinantal point processes with $J$-Hermitian correlation kernels

Mathematics – Probability

Scientific paper

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Scientific paper

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 $X$, i.e., 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 $k^{(n)}(x_1,...,x_n)=\det [K(x_i,x_j)]_{i,j=1,...,n}\,$. 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 $K(x,y)=-\bar{K(y,x)}$ 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 $K(x,y)$.

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