Limit theory for geometric statistics of point processes having fast decay of correlations
Résumé
Let P be a simple, stationary, clustering point process on Rd in the sense that its correlation functions factorize up to an additive error decaying exponentially fast with the separation distance. Let Pn:=P∩Wn be its restriction to windows Wn:=[−n1/d2,n1/d2]d⊂Rd. We consider the statistic Hξn:=∑x∈Pnξ(x,Pn) where ξ(x,Pn) denotes a score function representing the interaction of x with respect to Pn. When ξ depends on local data in the sense that its radius of stabilization has an exponential tail, we establish expectation asymptotics, variance asymptotics, and central limit theorems for Hξn and, more generally, for statistics of the random measures μξn:=∑x∈Pnξ(x,Pn)δn−1/dx, as Wn↑Rd. This gives the limit theory for non-linear geometric statistics (such as clique counts, the number of Morse critical points, intrinsic volumes of the Boolean model, and total edge length of the k-nearest neighbor graph) of determinantal point processes with fast decreasing kernels, including the α-Ginibre ensembles, extending the Gaussian fluctuation results of Soshnikov [67] to non-linear statistics. It also gives the limit theory for geometric U-statistics of permanental point processes as well as the zero set of Gaussian entire functions, extending the central limit theorems of Nazarov and Sodin [50], which are also confined to linear statistics. The proof of the central limit theorem relies on a factorial moment expansion originating in [10,11] to show clustering of mixed moments of ξ. Clustering extends the cumulant method to the setting of purely atomic random measures, yielding the asymptotic normality of μξn.