An upper bound for $p_c$ in range-$R$ bond percolation in two and three dimensions

TL;DR

This paper establishes an upper bound for the critical probability in long-range bond percolation in two and three dimensions by linking it to epidemic models and analyzing branching random walk local times.

Contribution

It introduces a novel method connecting percolation thresholds with epidemic models and provides uniform bounds for branching random walk local times.

Findings

Derived an upper bound for $p_c$ in 2D and 3D percolation.

Connected percolation thresholds with SIR epidemic models.

Established uniform bounds for local times of branching random walks.

Abstract

An upper bound for the critical probability of long range bond percolation in $d=2$ and $d=3$ is obtained by connecting the bond percolation with the SIR epidemic model, thus complementing the lower bound result in Frei and Perkins arXiv:arch-ive/1603.04130. A key ingredient is that we establish a uniform bound for the local times of branching random walk by calculating their exponential moments and by using the discrete versions of Tanaka's formula and Garsia's Lemma.