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The probability of connection between two vertices cannot be monotone with the distance for Bernoulli Percolation on transitive graphs

Alberto M. Campos, Bernardo N. B. de Lima

TL;DR

The paper addresses whether the probability that two vertices are connected in a transitive graph under Bernoulli percolation decreases with distance. It shows transitivity alone does not guarantee monotonicity by constructing a transitive graph with non-monotone connection probabilities and introduces the Dust-Pipe percolation model, which exhibits a clear threshold behavior controlled by a parameter $\lambda$. A key result is the existence of a nontrivial critical value $\lambda_c(\mathbb{Z}^d)$ (and related $\tau_c$) that governs whether monotonicity holds above or below the threshold, highlighting a phase-transition-like phenomenon in a generalized continuous-percolation setting. The work confirms that monotonicity is not a universal property of transitive graphs and connects the discrete Bernoulli framework to a broader, geometry-aware model, offering insights into percolation on $\mathbb{Z}^d$ and related lattices.

Abstract

A popular question in Bernoulli percolation models is if the probability of connection between two vertices in a transitive graph decays monotonically with the distance between these two vertices. For example, on the square lattice is an open question to prove that the probability of the origin being connected to the vertex $(0,n)$ is monotone in $n$. In this short note, we exhibit an example of a transitive graph in which the probability of connection between vertices does not necessarily decay as the distance of those vertices grows. We also define a critical point for percolation in $\mathbb{Z}^d$, in which using a generalization of the percolation process it is possible to see the same phenomena happening in the embedding of $\mathbb{Z}^d$ over $\mathbb{R}^d$.

The probability of connection between two vertices cannot be monotone with the distance for Bernoulli Percolation on transitive graphs

TL;DR

The paper addresses whether the probability that two vertices are connected in a transitive graph under Bernoulli percolation decreases with distance. It shows transitivity alone does not guarantee monotonicity by constructing a transitive graph with non-monotone connection probabilities and introduces the Dust-Pipe percolation model, which exhibits a clear threshold behavior controlled by a parameter . A key result is the existence of a nontrivial critical value (and related ) that governs whether monotonicity holds above or below the threshold, highlighting a phase-transition-like phenomenon in a generalized continuous-percolation setting. The work confirms that monotonicity is not a universal property of transitive graphs and connects the discrete Bernoulli framework to a broader, geometry-aware model, offering insights into percolation on and related lattices.

Abstract

A popular question in Bernoulli percolation models is if the probability of connection between two vertices in a transitive graph decays monotonically with the distance between these two vertices. For example, on the square lattice is an open question to prove that the probability of the origin being connected to the vertex is monotone in . In this short note, we exhibit an example of a transitive graph in which the probability of connection between vertices does not necessarily decay as the distance of those vertices grows. We also define a critical point for percolation in , in which using a generalization of the percolation process it is possible to see the same phenomena happening in the embedding of over .
Paper Structure (3 sections, 7 theorems, 24 equations, 3 figures)

This paper contains 3 sections, 7 theorems, 24 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['teo:conterexample']}
  3. The Dust-Pipe percolation

Key Result

Theorem 1

For every $\beta \in (0,1)$, there exists a transitive graph $G$ and vertices $x, y \in \mathbb{V}(G)$ such that $\|x\|_G < \|y\|_G$, but $\mathbb{P}_p(o \longleftrightarrow x) < \mathbb{P}_p(o \longleftrightarrow y)$ for every $p \in [\beta,1)$.

Figures (3)

  • Figure 1: Representation of the graph $P$, and elements $e_i$, $i\in \{1,...,6\}$ responsible for assembling the final graph $G$.
  • Figure 2: A representation of $G_2$, with copies of graph $P$ glued together in a tree-like structure, representing a piece of graph $G$.
  • Figure 3: Representation of the graph $P_6$, and the closest paths between $o$ and $(1,0,0)$.

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Lemma 1
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 4 more