Bernoulli percolation is a model for random media introduced by Broadbent and Hammersley in 1957. In this process, each vertex of a given graph is occupied or vacant, with respective probabilities $p$ and $1-p$, independently of the other vertices (for some parameter $p$). It is arguably one of the simplest models from statistical mechanics displaying a phase transition as the parameter $p$ varies, i.e. a drastic change of behavior at some critical value $p_c$, and it has been widely studied.
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: for percolation on ${\mathbb Z}^3$ with parameter $p=1/2$, we prove that almost surely, all words can be embedded. We also discuss various extensions of this result. This talk is based on a joint work with Augusto Teixeira (IMPA) and Vincent Tassion (ETH Zürich).