Distance Functions, Critical Points, and the Topology of Random \v{C}ech Complexes

TL;DR

This paper analyzes the critical points of the distance function to a random point set in Euclidean space, revealing their limiting behavior and linking to the topology of random Čech complexes.

Contribution

It provides explicit formulas and limit theorems for the number of critical points of various Morse indices in random point sets, connecting to topological invariants.

Findings

Explicit expectations and variances for critical points as density grows

Distributional limit theorems for the number of critical points

Connections established between critical points and Betti numbers of Čech complexes

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 \v{C}ech complex based on $P$ were studied.