The tail of the crossing probability in near-critical percolation --- an appendix to Ahlberg & Steif [arXiv:1405.7144]

TL;DR

This paper determines the superexponentially small tail behavior of crossing probabilities in near-critical planar percolation, contrasting with the different tail decay in dynamical percolation, using scale covariance properties.

Contribution

It provides the first precise characterization of the tail decay in near-critical percolation, extending the understanding of crossing probabilities.

Findings

Tail probability decays superexponentially in near-critical percolation

Contrast with exponential or superpolynomial decay in dynamical percolation

Utilizes scale covariance to simplify the proof

Abstract

We answer a question of Ahlberg and Steif (2014) by finding the tail behaviour of the crossing probability in near-critical planar percolation. Interestingly, this superexponentially small behaviour is different from the case of dynamical percolation, where the analogous tail probability was proved to be at least exponential and at most superpolynomial by Hammond, Mossel and Pete (2012). The proof is simple, given the scale covariance established by Garban, Pete and Schramm (2013).