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

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

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Published 30 November 2014