Percolation of arbitrary words in {0,1}N
Let G be a (possibly directed) locally finite graph with countably infinite vertex set V. Assign to each vertex a 0 or a 1, with probability p or 1-p, respectively, and take all vertices independent of one another. We want to know which sequences of zeroes and ones occur (with positive probability) along some selfavoiding path on G. The traditional problem in (site) percolation is whether the sequence (1,1,1,…) occurs on some path starting at a fixed vertex v 0. So - called AB-percolation occurs if the sequence (1,0,1,0,1,0,…) occurs with positive probability on some path starting at v0. We concentrate here on the questions (a): whether (with positive probability) all words are seen from v0 and (b): whether all words are seen somewhere on G with probability 1. We also consider similar questions with "all words" replaced by "almost all words".