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Random chain complexes of real vector spaces

Ayat Ababneh, Matthew Kahle

TL;DR

This work introduces a natural real-vector-space model for random chain complexes and analyzes their homology. By conditioning on ensembles defined via rank-stratified varieties and exploiting a monotonicity principle, it derives precise Betti-number behavior for lengths $1$, $2$, and $3$, showing that the total Betti sum $\sum_i \beta_i$ equals $|\chi|$ almost surely under mild dimension constraints and equal-dimension settings. In particular, odd-length equal-dimension complexes are almost surely exact, while even-length cases exhibit an evenly distributed Betti-vector constrained by $|\chi|$, suggesting a general heuristic that nontrivial homology is rare unless forced by topology. The paper also lays out a flexible framework to study longer complexes, presents conjectures for the general length, and highlights computational and algebraic directions, including potential ties to torsion phenomena and Cohen–Lenstra-type statistics for random integer matrices.

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 $|χ|$ 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.

Random chain complexes of real vector spaces

TL;DR

This work introduces a natural real-vector-space model for random chain complexes and analyzes their homology. By conditioning on ensembles defined via rank-stratified varieties and exploiting a monotonicity principle, it derives precise Betti-number behavior for lengths , , and , showing that the total Betti sum equals almost surely under mild dimension constraints and equal-dimension settings. In particular, odd-length equal-dimension complexes are almost surely exact, while even-length cases exhibit an evenly distributed Betti-vector constrained by , suggesting a general heuristic that nontrivial homology is rare unless forced by topology. The paper also lays out a flexible framework to study longer complexes, presents conjectures for the general length, and highlights computational and algebraic directions, including potential ties to torsion phenomena and Cohen–Lenstra-type statistics for random integer matrices.

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 case. We are interested here in the homological properties of these random complexes. For chain complexes of length or , we characterize the Betti numbers almost surely, in terms of the dimensions of the vector spaces. We further examine complexes of length 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 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.
Paper Structure (10 sections, 9 theorems, 45 equations)

This paper contains 10 sections, 9 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. The model
  3. Conditioning on an event of probability zero
  4. Random chain complexes
  5. Random chain complexes of length $1$, $2$, and $3$
  6. Chain complexes of length 1
  7. Chain complexes of length 2
  8. Chain complexes of length 3
  9. Random complexes with vector spaces of equal dimension
  10. Questions

Key Result

Proposition 2.1

There exists a chain complex with $\dim A_i = a_i$ for $i = 0, \dots, n$ and $\hbox{rank}d_i=r_i$ for $i=1, \dots, n$, if and only if $r_i + r_{i+1} \le a_i$ for every $i = 0, \dots, n$.

Theorems & Definitions (16)

  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3: Monotonocity Lemma
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 6 more