Non-triviality in a totally asymmetric one-dimensional Boolean percolation model on a half-line

TL;DR

This paper explores a unique regime in a one-dimensional Boolean percolation model on a half-line where the entire volume is covered almost surely without forming an unbounded component, revealing a third percolation phase.

Contribution

It introduces a new percolation regime in a Boolean model with asymmetric grains on a half-line, providing explicit conditions for unbounded component existence.

Findings

No unbounded component exists almost surely in this regime.

The covered volume fraction is one despite the absence of an unbounded component.

An explicit criterion characterizes the presence of an unbounded occupied component.

Abstract

It is well known that there are two regimes in a standard one-dimensional Boolean percolation model: either the entire space is covered a.s., or the covered volume fraction is strictly less than one. The aim of this work is to demonstrate that there is a third possibility in a Boolean model with totally asymmetric grains on a half-line: a.s. there is no unbounded component, but the covered volume fraction is one. An explicit condition is given characterizing the existence of an unbounded occupied component.