Journal
HOMOLOGY HOMOTOPY AND APPLICATIONS
Volume 16, Issue 2, Pages 311-344Publisher
INT PRESS BOSTON, INC
DOI: 10.4310/HHA.2014.v16.n2.a18
Keywords
distance function; critical points; Morse index; Cech complex; Poisson process; central limit theorem; Betti numbers
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Funding
- Adams Fellowship Program of the Israel Academy of Sciences and Humanities
- TOPOSYS [FP7-ICT-318493-STREP]
- AFOSR [FA8655-11-1-3039]
- ERC Advanced Grant, URSAT [20120216]
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For a finite set of points P in R-d, the function d(p): R-d -> 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 [16, 17] in which the Betti numbers of the random Cech complex based on P were studied.
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