Abstract
We consider supercritical vertex percolation in \(\mathbb{Z}^d \)d with any non-degenerate uniform oriented pattern of connection. In particular, our results apply to the more special unoriented case. We estimate the probability that a large region is isolated from ∞. In particular, “pancakes” with a radius r→∞ and constant thickness, parallel to a constant linear subspace L, are isolated with probability, whose logarithm grows asymptotically as ≍rdim(L) if percolation is possible across L and as ≍rdim(L)−1 otherwise. Also we estimate probabilities of large deviations in invariant measures of some cellular automata.
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