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The clairvoyant demon has a hard task. (English) Zbl 0974.60091
Summary: Consider the integer lattice \(L= \mathbb{Z}^2\). For some \(m\geq 4\), let us colour each column of this lattice independently and uniformly with one of \(m\) colours. We do the same for the rows, independently of the columns. A point of \(L\) will be called blocked if its row and column have the same colour. 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, avoiding the blocked points. As a problem arising in distributed computing, it has been conjectured that for \(m\geq 4\) the configuration percolates with positive probability. This question remains open, but 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.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60C05 Combinatorial probability
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