Homology, Homotopy and Applications

Volume 16 (2014)

Number 2

Distance functions, critical points, and the topology of random Čech complexes

Pages: 311 – 344

DOI: https://dx.doi.org/10.4310/HHA.2014.v16.n2.a18

Authors

Omer Bobrowski (Department of Mathematics, Duke University, Durham, North Carolina, U.S.A.)

Robert J. Adler (Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa, Israel)

Abstract

For a finite set of points $\mathcal{P}$ in $\mathbb{R}^d$, the function $d_{\mathcal{P}} : \mathbb{R}^d \to \mathbb{R}^+$ measures Euclidean distance to the set $\mathcal{P}$. We study the number of critical points of $d_{\mathcal{P}}$ when $\mathcal{P}$ is a Poisson process. In particular, we study the limit behavior of $N_k$—the number of critical points of $d_{\mathcal{P}}$ 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 [16, 17] in which the Betti numbers of the random Čech complex based on $\mathcal{P}$ were studied.

Keywords

distance function, critical points, Morse index, Čech complex, Poisson process, central limit theorem, Betti numbers

2010 Mathematics Subject Classification

55U10, 58K05, 60D05, 60F05, 60G55

Published 30 November 2014