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

OMER BOBROWSKI; ROBERT ADLER

doi:10.4310/HHA.2014.v16.n2.a18

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

OMER BOBROWSKI, ROBERT ADLER

Technion - Israel Institute of Technology

Research output: Contribution to journal › Article › peer-review

Abstract

For a finite set of points P in Rd, 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 Čech complex based on P were studied.

Original language	English
Pages (from-to)	311-344
Number of pages	34
Journal	Homology, Homotopy and Applications
Volume	16
Issue number	2
DOIs	https://doi.org/10.4310/HHA.2014.v16.n2.a18
State	Published - 2014

Keywords

Betti numbers
Central limit theorem
Critical points
Distance function
Morse index
Poisson process
Čech complex

All Science Journal Classification (ASJC) codes

Mathematics (miscellaneous)

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10.4310/HHA.2014.v16.n2.a18

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title = "Distance functions, critical points, and the topology of random {\v C}ech complexes",

abstract = "For a finite set of points P in Rd, the function dP: ℝd → ℝ+ measures Euclidean distance to the set P. We study the number of critical points of dP when P is a Poisson process. In particular, we study the limit behavior of Nk-the number of critical points of dP with Morse index k-as the density of points grows. We present explicit computations for the normalized limiting expectations and variances of the Nk, as well as distributional limit theorems.We link these results to recent results in [16, 17] in which the Betti numbers of the random {\v C}ech complex based on P were studied.",

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AU - BOBROWSKI, OMER

AU - ADLER, ROBERT

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AB - For a finite set of points P in Rd, the function dP: ℝd → ℝ+ measures Euclidean distance to the set P. We study the number of critical points of dP when P is a Poisson process. In particular, we study the limit behavior of Nk-the number of critical points of dP with Morse index k-as the density of points grows. We present explicit computations for the normalized limiting expectations and variances of the Nk, 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 P were studied.

KW - Betti numbers

KW - Central limit theorem

KW - Critical points

KW - Distance function

KW - Morse index

KW - Poisson process

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DO - 10.4310/HHA.2014.v16.n2.a18

M3 - مقالة

SN - 1532-0073

VL - 16

SP - 311

EP - 344

JO - Homology, Homotopy and Applications

JF - Homology, Homotopy and Applications

IS - 2

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

Abstract

Keywords

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