Brochette percolation

TL;DR

This paper investigates a modified bond percolation model on the square lattice with inhomogeneous vertical columns, showing that percolation can occur even when horizontal edges are less likely to percolate than the critical threshold for homogeneous models.

Contribution

It introduces a new inhomogeneous percolation model with randomly selected vertical columns and demonstrates percolation persistence below the standard critical threshold.

Findings

Percolation occurs with positive probability when vertical columns are above critical threshold.

Horizontal edges can be less likely to percolate than the homogeneous critical point.

Inhomogeneity allows percolation at lower overall probabilities.

Abstract

We study bond percolation on the square lattice with one-dimensional inhomogeneities. Inhomogeneities are introduced in the following way: A vertical column on the square lattice is the set of vertical edges that project to the same vertex on $\mathbb{Z}$. Select vertical columns at random independently with a given positive probability. Keep (respectively remove) vertical edges in the selected columns, with probability $p$, (respectively $1-p$). All horizontal edges and vertical edges lying in unselected columns are kept (respectively removed) with probability $q$, (respectively $1-q$). We show that, if $p > p_c(\mathbb{Z}^2)$ (the critical point for homogeneous Bernoulli bond percolation) then $q$ can be taken strictly smaller then $p_c(\mathbb{Z}^2)$ in such a way that the probability that the origin percolates is still positive.