Loop percolation on discrete half-plane

TL;DR

This paper investigates the percolation properties of the loop soup on the discrete half-plane, establishing the critical intensity for percolation and demonstrating that large loops suffice for percolation in the supercritical regime.

Contribution

It precisely determines the critical intensity for percolation in the loop soup on the discrete half-plane and shows percolation persists with only large loops above a certain size.

Findings

Critical percolation threshold is at intensity 1/2.

Percolation occurs with only large loops in the supercritical regime.

Percolation does not occur at the critical intensity.

Abstract

We consider the random walk loop soup on the discrete half-plane and study the percolation problem, i.e. the existence of an infinite cluster of loops. We show that the critical value of the intensity is equal to 1/2. The absence of percolation at intensity 1/2 was shown in a previous work. We also show that in the supercritical regime, one can keep only the loops up to some large enough upper bound on the diameter and still have percolation.