Critical percolation on slabs with random columnar disorder

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 $\ddot{I}•>1$. 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 $\ddot{I}•$ (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.

28 pages, 11 figures