Distance Functions, Critical Points, and Topology for Some Random Complexes

Details Distance Functions, Critical Points, and Topology for Some Random Complexes Distance Functions, Critical Points, and Topology for Some Random Complexes

2011-07-24

arxiv.org/abs/1107.4775v1

Mathematics

Probability

50 pages, 3 figures

Scientific paper

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 random. In particular, we study the limit behavior of $N_k$ - the number of critical points of $d_P$ with Morse index $k$ - as the number of points in $P$ goes to infinity. 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 Kahle (2009), and Kahle and Meckes (2011), in which the Betti numbers of the random Cech complex based on $P$ were studied.

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