On Large Isolated Regions in Supercritical Percolation

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 ≍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.