No exceptional words for Bernoulli percolation

TL;DR

This paper proves that in three or more dimensions, with certain percolation probabilities, all infinite binary sequences are almost surely embedded in Bernoulli percolation configurations, extending previous results and providing explicit probability estimates.

Contribution

It resolves an open problem by showing all words are seen in Bernoulli percolation on Z^d for d ≥ 3 and p in a specific interval, extending to slabs and providing probability bounds.

Findings

All words are almost surely seen in site percolation on Z^d for d ≥ 3.

The result holds for p in (p_c(Z^d), 1 - p_c(Z^d)).

Explicit probability estimates are provided for finite boxes.

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.