On Large Isolated Regions in Supercritical Percolation

Author: Toom A.  

Publisher: Springer Publishing Company

ISSN: 0022-4715

Source: Journal of Statistical Physics, Vol.109, Iss.5-6, 2002-12, pp. : 1091-1108

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Abstract

We consider supercritical vertex percolation in {Bbb Z}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.