The Clairvoyant Demon Has a Hard Task

Peter Gacs

Published 2003-02-06Version 1

For some m \ge 4, let us color each column of the integer lattice L = Z^2 independently and uniformly into one of m colors. We do the same for the rows, independently from the columns. A point of L will be called blocked if its row and column have the same color. We say that this random configuration percolates if there is a path in L starting at the origin, consisting of rightward and upward unit steps, and avoiding the blocked points. As a problem arising in distributed computing, it has been conjectured that for m \ge 4, the configuration percolates with positive probability. This has now been proved (in a later paper), for large m. Here, we prove that the probability that there is percolation to distance n but not to infinity is not exponentially small in n. This narrows the range of methods available for proving the conjecture.