Simplicial Homology of Random Configurations

TL;DR

This paper analyzes the topology of random geometric complexes formed by Poisson points on a torus, deriving moments, distributional limits, and bounds for topological invariants like Betti numbers and Euler characteristic.

Contribution

It provides explicit calculations of moments, distributional convergence results, and concentration bounds for topological features of Poisson-based simplicial complexes.

Findings

Number of k-simplices' moments computed explicitly

Euler characteristic's mean and variance derived

Connected complex count converges to Gaussian law

Abstract

Given a Poisson process on a $d$-dimensional torus, its random geometric simplicial complex is the complex whose vertices are the points of the Poisson process and simplices are given by the \u{C}ech complex associated to the coverage of each point. By means of Malliavin calculus, we compute explicitly the n$th$ order moment of the number of $k$-simplices. The two first order moments of this quantity allow us to find the mean and the variance of the Euler caracteristic. Also, we show that the number of any connected geometric simplicial complex converges to the Gaussian law when the intensity of the Poisson point process tends to infinity. We use a concentration inequality to find bounds for the for the distribution of the Betti number of first order and the Euler characteristic in such simplicial complex.