Percolation of arbitrary words in one dimension
  We consider a type of long-range percolation problem on the positive integers, motivated by earlier work of others on the appearance of (in)finite words within a site percolation model. The main issue is whether a given infinite binary word appears within an iid Bernoulli sequence at locations that satisfy certain constraints. We settle the issue in some cases, and provide partial results in others. 
Geoffrey R. Grimmett, Thomas M. Liggett and Thomas Richthammer