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Upper bounds for critical probabilities in Bernoulli Percolation models

Pablo A. Gomes, Alan Pereira, Remy Sanchis

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 $d \geq 3$.

Upper bounds for critical probabilities in Bernoulli Percolation models

Abstract

We consider bond and site Bernoulli Percolation in both the oriented and the non-oriented cases on and obtain rigorous upper bounds for the critical points in those models for every dimension .

Paper Structure

This paper contains 11 sections, 9 theorems, 11 equations, 1 table.

Table of Contents

  1. Introduction
  2. The models and main results
  3. Bond percolation
  4. Site percolation
  5. Explicit bounds
  6. The Dynamical Couplings
  7. Proof of Theorems
  8. Proof of Theorem \ref{['theo:bounds']}
  9. Proof of Theorem \ref{['theo:bondO']}
  10. Proof of Theorem \ref{['theo:siteNO']}
  11. Proof of Theorem \ref{['theo:siteO']}

Key Result

Theorem 2.1

Consider non-oriented bond Bernoulli percolation and let $p^{\ast}(d)$ be the unique solution in $(0,1)$ of Then, for every $d\geq 3$, we have $p_c^b(d) \leq p^{\ast}(d)$.

Theorems & Definitions (13)

  • Definition 1
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.3
  • Proposition 3.1
  • Proposition 3.2
  • ...and 3 more