Random Chain Complexes

TL;DR

This paper investigates the properties of random finite-dimensional chain complexes over finite fields, revealing that most have minimal homology, with detailed probabilistic behavior depending on the field size and complex dimension.

Contribution

It provides a probabilistic analysis of homology in random chain complexes, showing the likelihood of minimal homology and its asymptotic behavior as field size and complex dimension vary.

Findings

Most complexes have zero or one-dimensional homology.

Probability concentrates on minimal homology as field size increases.

Distribution of homology dimension decreases super-exponentially with increasing complex dimension.

Abstract

We study random, finite-dimensional, ungraded chain complexes over a finite field and show that for a uniformly distributed differential a complex has the smallest possible homology with the highest probability: either zero or one-dimensional homology depending on the parity of the dimension of the complex. We prove that as the order of the field goes to infinity the probability distribution concentrates in the smallest possible dimension of the homology. On the other hand, the limit probability distribution, as the dimension of the complex goes to infinity, is a super-exponentially decreasing, but strictly positive, function of the dimension of the homology.