Distance Functions, Critical Points, and the Topology of Random Čech Complexes

Omer Bobrowski; Robert J. Adler

paper

Distance Functions, Critical Points, and the Topology of Random Čech Complexes

arXiv:1107.4775

Abstract

For a finite set of points $P$ in $R^{d}$ , the function $d_{P} : R^{d} \to R^{+}$ measures Euclidean distance to the set $P$ . We study the number of critical points of $d_{P}$ when $P$ is a Poisson process. In particular, we study the limit behavior of $N_{k}$ - the number of critical points of $d_{P}$ with Morse index $k$ - as the density of points grows. We present explicit computations for the normalized, limiting, expectations and variances of the $N_{k}$ , as well as distributional limit theorems. We link these results to recent results in which the Betti numbers of the random Čech complex based on $P$ were studied.