Random chain complexes of real vector spaces

TL;DR

This paper studies the homological properties of random chain complexes of real vector spaces, showing that Betti numbers are almost surely minimal, revealing a tendency for nontrivial homology to be rare unless explicitly induced.

Contribution

It introduces a natural class of random chain complex models and characterizes their Betti numbers, demonstrating minimal homology in various finite-length settings.

Findings

Betti numbers are almost surely minimal across models

Sum of Betti numbers attains the trivial lower bound

Nontrivial homology is rare unless forced

Abstract

We introduce a natural class of models of random chain complexes of real vector spaces that some classical ensembles of random matrices, the length $1$ case. We are interested here in the homological properties of these random complexes. For chain complexes of length $1$ or $2$, we characterize the Betti numbers almost surely, in terms of the dimensions of the vector spaces. We further examine complexes of length $3$ with some constraints on dimensions, as well as complexes of arbitrary finite length in which all vector spaces have equal dimension. Across all these settings, we show that the sum of the Betti numbers is almost surely as small as possible, attaining a trivial lower bound $|\chi|$ dictated by the dimensions of the underlying vector spaces and the Euler formula. These results suggest an underlying algebraic heuristic for a phenomenon frequently observed in stochastic topology, that nontrivial homology rarely appears unless forced to.