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

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 $\mathbb{R}^d$.

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