Limit theorems for random cubical homology

2016/12/27 by Yasuaki Hiraoka, Hiraoka, Yasuaki, Kenkichi Tsunoda +1
Computer Science · Mathematics · #Algebraic Topology (math.AT) #FOS: Mathematics #Geometry and complex manifolds #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1612.08485

openalex publication_date 2016/12/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Tutte polynomials and random-cluster models in Bernoulli cell complexes
• Limit theorems for persistence diagrams

Related

• Limit theorems for Betti numbers of random simplicial complexes
• Large deviation principle for persistence diagrams of random cubical filtrations
• Homotopy Types of Random Cubical Complexes
• Betti numbers in the Random Connection Model for higher-dimensional simplicial complexes and the Boolean model
• Topology of random 2-dimensional cubical complexes