A lower bound for $p_c$ in range-$R$ bond percolation in two and three dimensions

TL;DR

This paper establishes lower bounds for the critical percolation probability in 2D and 3D range-$R$ bond percolation models, connecting epidemic models to percolation theory, and refines existing asymptotic conjectures.

Contribution

It introduces a novel method linking bond percolation with SIR epidemic models to derive lower bounds for critical probabilities in low dimensions.

Findings

Lower bounds match conjectured asymptotics for large range.

Refines previous bounds by Penrose.

Complements high-dimensional results by van der Hofstad and Sakai.

Abstract

We use the connection between bond percolation and SIR epidemics to establish lower bounds for the critical percolation probability in $2$ and $3$ dimensions as the range becomes large. The bound agrees with the conjectured asymptotics for the long range critical probability, refines results of M. Penrose, and complements results of van der Hofstad and Sakai in dimensions greater than $6$.