Critical percolation on slabs with random columnar disorder

TL;DR

This paper investigates a percolation model on slabs with inhomogeneous vertical columns determined by a renewal process, showing that percolation can occur at lower horizontal edge probabilities when vertical columns are more likely to be open.

Contribution

It demonstrates that in a slab with columnar disorder governed by a power-law renewal process, percolation persists at lower horizontal probabilities if vertical columns are sufficiently likely to be open.

Findings

Percolation occurs at lower horizontal probabilities when vertical columns are more likely to be open.

Large tail exponents in the renewal process ensure percolation at subcritical horizontal probabilities.

Vertical inhomogeneities influence the percolation threshold in slab models.

Abstract

We explore a bond percolation model on slabs $\mathbb{S}^+_k=\mathbb{Z}_+\times \mathbb{Z}_+\times\{0,\dots,k\}$ featuring one-dimensional inhomogeneities. In this context, a vertical column on the slab comprises the set of vertical edges projecting to the same vertex on $\mathbb{Z}_+\times\{0,\dots,k\}$. Columns are chosen based on the arrivals of a renewal process, where the tail distributions of inter-arrival times follow a power law with exponent $\phi>1$. Inhomogeneities are introduced as follows: vertical edges on selected columns are open (closed) with probability $q$ (respectively $1-q$), independently. Conversely, vertical edges within unselected columns and all horizontal edges are open (closed) with probability $p$ (respectively $1-p$). We prove that for all sufficiently large $\phi$ (depending solely on $k$), the following assertion holds: if $q>p_c(\mathbb{S}^+_k)$, then $p$ can be taken strictly smaller than $p_c(\mathbb{S}^+_k)$ in a manner that percolation still occurs.