Partial Isometries, Duality, and Determinantal Point Processes

A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures $\Xi$ on a space $S$ with measure $\lambda$, whose correlation functions are all given by determinants specified by an integral kernel $K$ called the correlation kernel. We consider a pair of Hilbert spaces, $H_{\ell}, \ell=1,2$, which are assumed to be realized as $L^2$-spaces, $L^2(S_{\ell}, \lambda_{\ell})$, $\ell=1,2$, and introduce a bounded linear operator ${\cal W} : H_1 \to H_2$ and its adjoint ${\cal W}^{\ast} : H_2 \to H_1$. We show that if ${\cal W}$ is a partial isometry of locally Hilbert--Schmidt class, then we have a unique DPP on $(\Xi_1, K_1, \lambda_1)$ associated with ${\cal W}^* {\cal W}$. In addition, if ${\cal W}^*$ is also of locally Hilbert--Schmidt class, then we have a unique pair of DPPs, $(\Xi_{\ell}, K_{\ell}, \lambda_{\ell})$, $\ell=1,2$. We also give a practical framework which makes ${\cal W}$ and ${\cal W}^{\ast}$ satisfy the above conditions. Our framework to construct pairs of DPPs implies useful duality relations between DPPs making pairs. For a correlation kernel of a given DPP our formula can provide plural different expressions, which reveal different aspects of the DPP. In order to demonstrate these advantages of our framework as well as to show that the class of DPPs obtained by this method is large enough to study universal structures in a variety of DPPs, we report plenty of examples of DPPs in one-, two-, and higher-dimensional spaces $S$, where several types of weak convergence from finite DPPs to infinite DPPs are given. One-parameter ($d \in \mathbb{N}$) series of infinite DPPs on $S=\mathbb{R}^d$ and $\mathbb{C}^d$ are discussed, which we call the Euclidean and the Heisenberg families of DPPs, respectively, following the terminologies of Zelditch.