On Large Isolated Regions in Supercritical Percolation.

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 ≍ r^{dim( L)} if percolation is possible across L and as ≍ r^{dim( L)−1} otherwise. Also we estimate probabilities of large deviations in invariant measures of some cellular automata. [ABSTRACT FROM AUTHOR]