Homological Percolation: The Formation of Giant k-Cycles

TL;DR

This paper extends the concept of giant components in percolation to higher-dimensional cycles using algebraic topology, analyzing their emergence and thresholds in continuum percolation models on tori.

Contribution

It introduces the notion of giant k-cycles in continuum percolation and characterizes their emergence thresholds and decay properties.

Findings

Giant k-cycles appear in the thermodynamic limit regime.

Thresholds for giant k-cycle emergence increase with k.

Probabilities of giant cycles decay exponentially.

Abstract

In this paper we introduce and study a higher-dimensional analogue of the giant component in continuum percolation. Using the language of algebraic topology, we define the notion of giant k-dimensional cycles (with 0-cycles being connected components). Considering a continuum percolation model in the flat d-dimensional torus, we show that all the giant k-cycles (k=1,...,d-1) appear in the regime known as the thermodynamic limit. We also prove that the thresholds for the emergence of the giant k-cycles are increasing in k and are tightly related to the critical values in continuum percolation. Finally, we provide bounds for the exponential decay of the probabilities of giant cycles appearing.