A remark on the convergence of Betti numbers in the thermodynamic regime

TL;DR

This paper proves that the expected Betti numbers of Čech complexes constructed from binomial point processes converge in the thermodynamic regime, advancing understanding of topological properties in large random structures.

Contribution

It establishes the convergence of Betti number expectations for Čech complexes on binomial point processes in the thermodynamic limit, a novel result in stochastic topology.

Findings

Expected Betti numbers converge in the thermodynamic regime

Provides theoretical foundation for topological analysis of large random complexes

Advances stochastic topology understanding in high-density regimes

Abstract

The convergence of the expectations of Betti numbers of \v{C}ech complexes built on binomial point processes in the thermodynamic regime is established.