No exceptional words for Bernoulli percolation
Probability
2019-11-13 v1
Abstract
Benjamini and Kesten introduced in 1995 the problem of embedding infinite binary sequences into a Bernoulli percolation configuration, known as "percolation of words". We give a positive answer to their Open Problem 2: almost surely, all words are seen for site percolation on Z^3 with parameter p = 1/2. We also extend this result in various directions, proving the same result for any dimension d at least three and for any value p in the interval (p_c(Z^d), 1 - p_c(Z^d)), and for restrictions to slabs. Finally, we provide an explicit estimate on the probability to find all words starting from a finite box.
Keywords
Cite
@article{arxiv.1911.04816,
title = {No exceptional words for Bernoulli percolation},
author = {Pierre Nolin and Vincent Tassion and Augusto Teixeira},
journal= {arXiv preprint arXiv:1911.04816},
year = {2019}
}
Comments
24 pages, 9 figures