Monotonicity of the critical point in two-dimensional oriented percolation with enhancement

TL;DR

This paper proves that adding diagonal edges with probability  in a 2D oriented percolation model decreases the critical percolation threshold, demonstrating a monotonic relationship.

Contribution

It establishes the strict monotonicity of the critical point in a 2D oriented percolation model with enhanced diagonal edges.

Findings

Critical parameter decreases as diagonal edge probability increases

Monotonicity of the critical point is rigorously proven

Enhancement influences percolation threshold in a predictable way

Abstract

In this note, we investigate Bernoulli oriented bond percolation with parameter $p$ on $\mathbb{Z}^2$. In addition to the standard edges, which are open with probability $p$, we introduce diagonal edges each open with probability $\varepsilon$. Every edge is open or closed independently of all other edges. We prove that the critical parameter for this model is strictly decreasing in $\varepsilon$.