Crossing probabilities for planar percolation

2020/11/09 by Laurin Köhler-Schindler, Köhler-Schindler, Laurin, Vincent Tassion +1 · 3 citations
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #68Q87 #Bayesian Methods and Mixture Models #Diffusion and Search Dynamics #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #primary: 60K35 #secondary: 82B43

paper · pdf · doi:10.48550/arxiv.2011.04618

openalex publication_date 2020/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a general Russo-Seymour-Welsh result valid for any invariant planar percolation process satisfying positive association. This means that the probability of crossing a rectangle in the long direction is related by a homeomorphism to the probability of crossing it in the short direction. This homeomorphism is universal in the sense that it depends only on the aspect ratio of the rectangle, and is uniform in the scale and the considered model.

Cited by

• Dropout Neural Network Training Viewed from a Percolation Perspective
• On the Continuity of Enhancement Percolation
• Planar percolation and the loop O(n) model

Related

• Crossing probabilities for Voronoi percolation
• The tail of the crossing probability in near-critical percolation --- an appendix to Ahlberg & Steif [arXiv:1405.7144]
• Universal crossing probability in anisotropic systems
• Crossing probabilities for critical Bernoulli percolation on slabs
• Universal crossing probabilities and incipient spanning clusters in directed percolation