Limit theorems for random cubical homology

TL;DR

This paper establishes limit theorems such as the law of large numbers and central limit theorem for Betti numbers and lifetime sums in random cubical sets, advancing understanding of their probabilistic topological properties.

Contribution

It introduces a framework for analyzing the asymptotic behavior of topological invariants in random cubical complexes, including new limit theorems and positivity results.

Findings

Law of large numbers for Betti numbers

Central limit theorem for lifetime sums

Positivity of limiting Betti numbers

Abstract

This paper studies random cubical sets in $\mathbb{R}^d$. Given a cubical set $X\subset \mathbb{R}^d$, a random variable $\omega_Q\in[0,1]$ 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 $\omega_Q\leq t$ for each $t\in[0,1]$. 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.