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Percolation threshold for metric graph loop soup

Yinshan Chang, Hang Du, Xinyi Li

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 $1/2$.

Percolation threshold for metric graph loop soup

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 , ) is .
Paper Structure (3 sections, 6 theorems, 28 equations, 1 figure)

This paper contains 3 sections, 6 theorems, 28 equations, 1 figure.

Table of Contents

  1. Introduction and the main result
  2. Preliminaries
  3. Proof of the main result

Key Result

Theorem 1

For any quasi-transitive transient metric graph ${\cal G}$ and any $x_o\in {\cal G}$, it holds that

Figures (1)

  • Figure 1: A possible realization of $K_n$, $C_n$, $\partial_i K_n$, $\partial^2_i K_n$ and an exemplary pivotal 2-bond. Note that in this figure $B_n$ and $C_n$ are only partially depicted.

Theorems & Definitions (15)

  • Theorem 1
  • Remark 2
  • Definition 1: Metric graphs
  • Definition 2: The metric graph loop soup
  • Proposition 3: No percolation at intensity $1/2$
  • proof
  • Proposition 4: Killed two-point function
  • Proposition 5: Russo's formula
  • proof
  • Proposition 6: FKG inequality
  • ...and 5 more