The Clairvoyant Demon Has A Hard Task

Consider the integer lattice L = Z2. For some m ≥ 4, let us color each column of this lattice 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 ≥ 4, the configuration percolates with positive probability. This has now been proved (in a later paper) for large m. 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. 1. STATEMENT OF THE RESULT 1.