Percolation threshold for metric graph loop soup

TL;DR

This paper establishes that the percolation threshold for metric graph loop soup on a broad class of transient graphs, including high-dimensional integer lattices, is exactly 1/2, advancing understanding of phase transitions in such models.

Contribution

It proves that the critical percolation threshold for metric graph loop soup on certain transient graphs is exactly 1/2, a significant theoretical result.

Findings

Percolation threshold is 1/2 for a large class of graphs.

Includes quasi-transitive graphs like ^d for d.

Advances understanding of phase transitions in loop soup models.

Abstract

In this short note, we show that the critical threshold for the percolation of metric graph loop soup on a large class of transient metric graphs (including quasi-transitive graphs such as $\mathbb{Z}^d$, $d\geq 3$) is $1/2$.