On percolation in a generalized backbend process

TL;DR

This paper extends the concept of backbend in oriented bond percolation by introducing a sequence-based generalization, analyzing critical probabilities on BCC lattices across various geometries, and establishing conditions for non-percolation at criticality.

Contribution

It introduces a generalized backbend framework for percolation, analyzing its impact on critical probabilities and percolation behavior on BCC lattices in different spatial domains.

Findings

Derived relationships between critical probabilities for various backbend sequences.

Established conditions under which no percolation occurs at critical probabilities.

Analyzed percolation thresholds on half-space, full-space, and slabs of the BCC lattice.

Abstract

We have generalized the idea of backbend in a nearest-neighbor oriented bond percolation process by considering a backbend sequence $\beta : \mathbb{Z}_+ \to \mathbb{Z}_+ \cup \{\infty\}$, and defining a $\beta$-backbend path from the origin as a path that never retreats further than $\beta(h)$ levels back from its record level $h$. We study the relationship between the critical probabilities of different percolation processes based on different backbend sequences on half-space, full-space, and half-slabs of the $d$-dimensional ($d \geq 2$) body-centered cubic (BCC) lattice. We also give sufficient conditions on the backbend sequences such that there will be no percolation at the critical probabilities of the corresponding percolation processes on half-space and full-space of the BCC lattice.