Check out the New York Sun's crossword for today, September 6, 2007.
The theme of this puzzle is "FROGGER", as we're told by the entry in the center; the crossword was constructed so that you can get from a square at the bottom to a square at the top going via only squares that contain letters in the word FROGGER. A picture of the solved crossword with just the FROGGER letters is at the left.
As it turned out, I'd been absent-mindedly flipping through Percolation by Geoffrey Grimmett right before taking a break to do this crossword. One of the standard results in the theory of percolationis that if we start with an infinite grid and fill in some proportion p of the squares at random, then with probability 1 there will be an infinitely large filled cluster containing the origin when p > 1/2, and if p < 1/2 there's an infinite filled cluster containing the origin with probability 0. (Usually one talks about open and closed bonds -- the lines connecting the sites -- instead of open and closed sites, but since the square lattice is self-dual that doesn't matter.)
So it's noteworthy that this is possible here; it wouldn't be possible in a random crossword. We wouldn't be able to get "far away" from a given site using only letters that are taken from some small set. The transition isn't going to be quite so sharp in the non-infinite cases, and the standard 15-by-15 crossword isn't all that close to infinity. Furthermore, finding a long path is even less likely in a crossword than in an ordinary square lattice, because in a crossword there are some black squares.
Furthermore, percolation theory usually assumes that sites are independent; this isn't true in a crossword, because the letters in sites adjacent to each other are correlated. For example, if a given square contains a T, the letter to the right of it or below it is probably more likely to be an H than otherwise, because the pair of letters "TH" is quite common. Any serious attempt to think about percolation in crosswords -- although I can't imagine anyone would study that seriously -- would have to take this into account.
A brief explanation, What is... Percolation by Harry Kesten, was published in May 2006 in the notices of the AMS.