Partial Isometries, Duality, and Determinantal Point Processes

Makoto Katori, Tomoyuki Shirai

Published 2019-03-12Version 1

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 prove that if both of ${\cal W}$ and ${\cal W}^{\ast}$ are partial isometries and both of ${\cal W}^{\ast} {\cal W}$ and ${\cal W} {\cal W}^{\ast}$ are of locally trace class, then we have unique pair of DPPs, $(\Xi_{\ell}, K_{\ell}, \lambda_{\ell})$, $\ell=1,2$, which satisfy useful duality relations. We assume that ${\cal W}$ admits an integral kernel $W$ on $L^2(S_1, \lambda_1)$, and give practical setting of $W$ which makes ${\cal W}$ and ${\cal W}^{\ast}$ satisfy the above conditions. In order to demonstrate that the class of DPPs obtained by our method is large enough to study universal structures in a variety of DPPs, we show many 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.