A Peierls bound for planar soft-stick percolation

Source abstract

In planar soft-stick percolation, each vertex of the square lattice independently opens one uniformly chosen outgoing arrow and, with probability $ε$, the opposite arrow as well. Motivated by the question on its critical parameter raised by Bäumler et al., we prove that $ε_{c}\le0.99<1$, establishing percolation below the two-arrow endpoint. More precisely, at $ε=0.99$ the origin has an infinite forward cluster with probability greater than 11/20. The proof uses a Peierls argument in which boundary turns are encoded by a three-state transfer matrix to bound the contour sum.