Continuum percolation for Gaussian zeroes and Ginibre eigenvalues (Q341491)

Consider a simple point process in the Euclidean plane. Given the realization of this process and a given \(r >0\), two points are said to be connected and are part of a cluster if there exists a sequence of points such that circles of radius \(r\) with them as centers form a continuum (that is, a Boolean percolation). For a given \(r\), the question addressed here is about the probability of observing infinite clusters in two special negatively dependent point processes, namely the \textit{J. Ginibre} ensemble (GE) [J. Math. Phys. 6, 440--449 (1965; Zbl 0127.39304)], and the Gaussian analytical function (GAF) zero process (described in [\textit{J. B. Hough} et al., Zeros of Gaussian analytic functions and determinantal point processes. Providence, RI: American Mathematical Society (AMS) (2009; Zbl 1190.60038)]). If there exists an \(r_c \in (0,\infty)\) such that the probability of observing infinite clusters is zero if \(r <r_c\) and is strictly positive if \(r > r_c\), then \(r_c\) is called the critical radius. In [Adv. Appl. Probab. 46, No. 1, 1--20 (2014; Zbl 1295.60059)], \textit{B. Błaszczyszyn} and \textit{D. Yogeshwaran} established the existence of a critical radius \(r_c\) for the GE; it is shown here that when \(r >r_c\), almost surely there is exactly one infinite cluster. For the GAF zero process, the paper establishes the existence of a critical radius \(r_c\), and also shows that when \(r >r_c\), almost surely there is exactly one infinite cluster (as in the case of GE).