Limit Theorems for Random Cubical Homology

Yasuaki Hiraoka; Kenkichi Tsunoda

doi:10.1007/s00454-018-0007-z

Limit Theorems for Random Cubical Homology

Yasuaki Hiraoka, Kenkichi Tsunoda

Advanced Institute for Materials Research (AIMR)

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

4 Citations (Scopus)

Abstract

This paper studies random cubical sets in R^d. Given a cubical set X⊂ R^d, a random variable ω_Q∈ [ 0 , 1 ] is assigned for each elementary cube Q in X, and a random cubical set X(t) is defined by the sublevel set of X consisting of elementary cubes with ω_Q≤ t for each t∈ [ 0 , 1 ]. Under this setting, the main results of this paper show the limit theorems (law of large numbers and central limit theorem) for Betti numbers and lifetime sums of random cubical sets and filtrations. In addition to the limit theorems, the positivity of the limiting Betti numbers is also shown.

Original language	English
Pages (from-to)	665-687
Number of pages	23
Journal	Discrete and Computational Geometry
Volume	60
Issue number	3
DOIs	https://doi.org/10.1007/s00454-018-0007-z
Publication status	Published - 2018 Oct 1

Keywords

Betti number
Cubical complex
Cubical homology
Random topology

ASJC Scopus subject areas

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

Access to Document

10.1007/s00454-018-0007-z

Cite this

@article{5b62d6886d0a49d8b43e12afcf8e4417,

title = "Limit Theorems for Random Cubical Homology",

abstract = "This paper studies random cubical sets in Rd. Given a cubical set X⊂ Rd, a random variable ωQ∈ [ 0 , 1 ] is assigned for each elementary cube Q in X, and a random cubical set X(t) is defined by the sublevel set of X consisting of elementary cubes with ωQ≤ t for each t∈ [ 0 , 1 ]. Under this setting, the main results of this paper show the limit theorems (law of large numbers and central limit theorem) for Betti numbers and lifetime sums of random cubical sets and filtrations. In addition to the limit theorems, the positivity of the limiting Betti numbers is also shown.",

keywords = "Betti number, Cubical complex, Cubical homology, Random topology",

author = "Yasuaki Hiraoka and Kenkichi Tsunoda",

note = "Funding Information: The authors are partly supported by JST CREST Mathematics (15656429). Publisher Copyright: {\textcopyright} 2018, Springer Science+Business Media, LLC, part of Springer Nature.",

year = "2018",

month = oct,

day = "1",

doi = "10.1007/s00454-018-0007-z",

language = "English",

volume = "60",

pages = "665--687",

journal = "Discrete and Computational Geometry",

issn = "0179-5376",

publisher = "Springer New York",

number = "3",

}

TY - JOUR

T1 - Limit Theorems for Random Cubical Homology

AU - Hiraoka, Yasuaki

AU - Tsunoda, Kenkichi

PY - 2018/10/1

Y1 - 2018/10/1

N2 - This paper studies random cubical sets in Rd. Given a cubical set X⊂ Rd, a random variable ωQ∈ [ 0 , 1 ] is assigned for each elementary cube Q in X, and a random cubical set X(t) is defined by the sublevel set of X consisting of elementary cubes with ωQ≤ t for each t∈ [ 0 , 1 ]. Under this setting, the main results of this paper show the limit theorems (law of large numbers and central limit theorem) for Betti numbers and lifetime sums of random cubical sets and filtrations. In addition to the limit theorems, the positivity of the limiting Betti numbers is also shown.

AB - This paper studies random cubical sets in Rd. Given a cubical set X⊂ Rd, a random variable ωQ∈ [ 0 , 1 ] is assigned for each elementary cube Q in X, and a random cubical set X(t) is defined by the sublevel set of X consisting of elementary cubes with ωQ≤ t for each t∈ [ 0 , 1 ]. Under this setting, the main results of this paper show the limit theorems (law of large numbers and central limit theorem) for Betti numbers and lifetime sums of random cubical sets and filtrations. In addition to the limit theorems, the positivity of the limiting Betti numbers is also shown.

KW - Betti number

KW - Cubical complex

KW - Cubical homology

KW - Random topology

UR - http://www.scopus.com/inward/record.url?scp=85046889012&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85046889012&partnerID=8YFLogxK

U2 - 10.1007/s00454-018-0007-z

DO - 10.1007/s00454-018-0007-z

M3 - Article

AN - SCOPUS:85046889012

VL - 60

SP - 665

EP - 687

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 3

ER -

Limit Theorems for Random Cubical Homology

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this