Limit theory for geometric statistics of clustering point processes

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 := [− 1/d 2 , n1/d 2 ] d ⊂ Rd. We consider the statistic H n := ∑ x∈Pn ξ(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 H n and, more generally, for statistics of the random measures μn := ∑ x∈Pn ξ(x,Pn)δn−1/dx, asWn ↑ R d. 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 having fast decreasing kernels, including the β-Ginibre ensembles, extending the Gaussian fluctuation results of Soshnikov [68] to non-linear statistics. It also gives the limit theory for geometric U-statistics of α-permanental point processes (for 1/α ∈ N), α-determinantal point processes (for −1/α ∈ N), as well as the zero set †Research supported in part by DST-INSPIRE faculty award and TOPOSYS Grant. ‡Research supported in part by NSF grant DMS-1406410