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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 εc≤0.99<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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Cite

BibTeXRIS
BibTeX
@misc{chen2026peierls,
  title = {{A Peierls bound for planar soft-stick percolation}},
  author = {Shitao Chen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10067},
  eprint = {2609.10067},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10067},
}
RIS
TY  - UNPB
TI  - A Peierls bound for planar soft-stick percolation
AU  - Shitao Chen
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.10067
UR  - https://arxiv.org/abs/2609.10067
AB  - 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≤0.99<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.
ER  -