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Section: New Results

Limit Theory for Geometric Statistics of Clustering Point Processes

Let P be a simple, stationary, clustering point process on the $d$-dimensional Euclidean space, in the sense that its correlation functions factorize up to an additive error decaying exponentially fast with the separation distance. Let $P_n$ be its restriction to a hypercube windows of volume $n$. We consider statistics of $P_n$ admitting the representation as sums of spatially dependent terms $H_n={\sum }_{x\in P_n}\xi (x,P_n)$, where $\xi (x,P_n)$ is a real valued (score) function, representing the interaction of $x$ with $P_n$. When the score function depends locally on $P_n$ in the sense that its radius of stabilization has an exponential tail, we establish expectation asymptotics, variance asymptotics, and central limit theorems for $H_n$ as the volume n of the window goes to infinity.

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 $\alpha$-Ginibre ensembles. It also gives the limit theory for geometric U-statistics of permanental point processes as well as the zero set of Gaussian entire functions. This extends the existing literature treating the limit theory of sums of stabilizing scores of Poisson and binomial input. In the setting of clustering point processes, it also extends the results of Soshnikov  [61] as well as work of Nazarov and Sodin  [55].

The proof of the central limit theorem relies on a factorial moment expansion originating in Blaszczyszyn  [34] to show clustering of mixed moments of the score function. Clustering extends the cumulant method to the setting of purely atomic random measures, yielding the asymptotic normality of $H_n$.