Percolation with random one-dimensional reinforcements

TL;DR

This paper investigates how inhomogeneous bond percolation on a product graph with a random one-dimensional reinforcement region affects percolation, establishing conditions for the persistence of non-percolative phases.

Contribution

It introduces a model with a random reinforcement region in inhomogeneous percolation and derives conditions based on the moments of the radii for non-percolation to persist.

Findings

Non-percolative phase persists for subcritical p and q<1.

Conditions depend on moments of the radii and growth of G.

Results extend understanding of inhomogeneous percolation with random reinforcements.

Abstract

We study inhomogeneous Bernoulli bond percolation on the graph $G \times \mathbb{Z}$, where $G$ is a connected quasi-transitive graph. The inhomogeneity is introduced through a random region $R$ around the origin axis $\{0\}\times\mathbb{Z}$, where each edge in $R$ is open with probability $q$ and all other edges are open with probability $p$. When the region $R$ is defined by stacking or overlapping boxes with random radii centered along the origin axis, we derive conditions on the moments of the radii, based on the growth properties of $G$, so that for any subcritical $p$ and any $q<1$, the non-percolative phase persists.