Extreme paths in oriented 2D Percolation

TL;DR

This paper proves a key result about the behavior of extreme paths in 2D oriented percolation, with implications for related models and percolation probabilities.

Contribution

It provides a rigorous proof of a previously unproven result about leftmost and rightmost paths in 2D bond percolation, extending its validity to related models.

Findings

Monotonicity in percolation probability between sites

Counter-intuitive correlation inequality

Applicability to discrete time contact process and site percolation

Abstract

A useful result about leftmost and rightmost paths in two dimensional bond percolation is proved. This result was introduced without proof in \cite{G} in the context of the contact process in continuous time. As discussed here, it also holds for several related models, including the discrete time contact process and two dimensional site percolation. Among the consequences are a natural monotonicity in the probability of percolation between different sites and a somewhat counter-intuitive correlation inequality.