Topological data analysis of continuum percolation with disks

TL;DR

This paper applies topological data analysis, specifically persistent homology, to study the topological features of continuum percolation with disks in 2D, revealing insights about the percolation transition.

Contribution

It introduces a novel application of persistent homology to continuum percolation, linking topological features to percolation phenomena.

Findings

Longest persistent invariant appears near the percolation transition

Topological features change systematically with disk radius and number

Persistent homology captures critical topological changes during percolation

Abstract

We study continuum percolation with disks, a variant of continuum percolation in two-dimensional Euclidean space, by applying tools from topological data analysis. We interpret each realization of continuum percolation with disks as a topological subspace of $[0,1]^2$ and investigate its topological features across many realizations. We apply persistent homology to investigate topological changes as we vary the number and radius of disks. We observe evidence that the longest persisting invariant is born at or near the percolation transition.