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arXiv · 2609.10067

A Peierls bound for planar soft-stick percolation

Shitao Chen
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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.

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Shitao Chen. 2026-09-09. A Peierls bound for planar soft-stick percolation. https://arxiv.org/abs/2609.10067

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