No Exceptional Words for Site Percolation on Z^3
City University of Hong Kong
Abstract :
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 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).