Limit theory for statistics of random geometric structures

Abstract:

Functionals of large geometric structures on finite input $X \subset {\Bbb R}^d$ often consist
of sums of spatially dependent terms admitting the representation
\begin{equation}
\label{basic}
\sum_{x \in X} \xi(x,X), \quad\quad\quad\quad\quad (1)
\end{equation}
where the ${\Bbb R}$-valued score function $X$, defined on pairs $(x, X)$, $x \in X$, represents the interaction of $x$ with respect to $X$. The sums (1) typically describe a global feature of the geometric structure in terms of a sum of local contributions $\xi(x,X)$, $x \in X$. A large number of functionals and statistics in spatial random systems may be cast in the form (1) for appropriately chosen $\xi$.
This includes total edge length and simplex counts in random graphs, global statistics of germ-grain models, functionals of convex hulls of random samples, total coverage in random sequential adsorption models, and statistics of spatial birth growth models.

If the input $X$ is a clustering point process and if the score function $\xi$ is determined by `local' data (i.e., stabilizing), then we establish general expectation and variance asymptotics as well as central limit theorems for the suitably scaled sums
\begin{equation}
\label{basic1}
\sum_{x \in X \cap W_n} \xi(x,X \cap W_n) \quad\quad\quad\quad\quad(2)
\end{equation}
as $W_n \uparrow {\Bbb R}^d$.
Given a score function $\xi$ which is determined by local data we thus obtain general limit theorems for the sums (2) when the input is given by a determinantal or permanental point process with a fast decreasing kernel (e.g. the Ginibre ensemble), the zero set of a Gaussian entire function, or rarified Gibbsian input. 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 work of Nazarov and Sodin as well as work of Soshnikov, which are confined to linear statistics $\sum_{x \in X \cap W_n} \xi(x)$.

The talk is based on joint work with B. Blaszczyszyn and D. Yogeshwaran.