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

TL;DR

This work proves that in two-dimensional oriented Bernoulli bond percolation augmented with static $\varepsilon$-open diagonal edges, the critical parameter $p_c(\varepsilon)$ decreases strictly with increasing $\varepsilon$. The authors analyze the asymptotic speed of the right edge $\alpha(p,\varepsilon)=\lim_n r(N^-_n)/n$, showing it equals zero at criticality via a Parallelogram-constructed $1$-dependent percolation argument. They then establish monotonicity of the critical point by coupling processes with different $\varepsilon$ values and using domination lemmas to sustain improvements in growth through time, effectively proving $p_c(\tilde{\varepsilon})<p_c(\varepsilon)$ whenever $0\le \varepsilon<\tilde{\varepsilon}\le 1$. The results extend monotonicity phenomena to oriented percolation with enhancements and rely on stochastic domination and careful path-wise couplings, building on prior work on oriented models and dynamic analogues. The findings imply enhancements strictly facilitate infinite connectivity in this oriented setting, with potential implications for related percolation models and their critical behavior.

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 $\varepsilon$.

Theorems & Definitions (10)

• Theorem 1
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Lemma 3.1
• Lemma 3.2
• proof : Proof of Theorem \ref{['thm:monotonicityepsilon']}.
• proof : Proof of Lemma \ref{['lemma:domination']}.
• proof : Proof of Lemma \ref{['lemma:coupling']}.