Upper bounds for critical probabilities in Bernoulli Percolation models

TL;DR

This paper establishes rigorous upper bounds for the critical probabilities in Bernoulli Percolation models on integer lattices across all dimensions three and higher, advancing understanding of phase transition thresholds.

Contribution

It provides the first comprehensive upper bounds for critical points in both oriented and non-oriented Bernoulli Percolation models for all dimensions d ≥ 3.

Findings

Upper bounds for critical probabilities in Bernoulli Percolation models

Applicable to both oriented and non-oriented cases

Valid for all dimensions d ≥ 3

Abstract

We consider bond and site Bernoulli Percolation in both the oriented and the non-oriented cases on $\mathbb{Z}^d$ and obtain rigorous upper bounds for the critical points in those models for every dimension $d \geq 3$.