Forward clusters for degenerate random environments

TL;DR

This paper investigates connectivity and asymptotic behavior in specific infinite directed graphs on Z^2, providing conditions for the absence of infinite components in the complement and improving bounds for percolation thresholds.

Contribution

It introduces new conditions for connectivity in degenerate random environments and refines the lower bounds for the critical probability in oriented site percolation.

Findings

Conditions for the absence of infinite components in the complement of C_o

Improved lower bound for the critical occupation probability in 2D oriented site percolation

Enhanced understanding of connectivity properties in directed random graphs

Abstract

We consider connectivity properties and asymptotic slopes for certain random directed graphs on $Z^2$ in which the set of points $C_o$ that the origin connects to is always infinite. We obtain conditions under which the complement of $C_o$ has no infinite connected component. Applying these results to one of the most interesting such models leads to an improved lower bound for the critical occupation probability for oriented site percolation on the triangular lattice in 2 dimensions.