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We prove several facts concerning Lipschitz percolation, including the following. The critical probability L for the existence of an open Lipschitz surface in site percolation on ℤ with ≥ 2 satisfies the improved bound L ≤ 1 − 1/[8( − 1)]. Whenever > L, the height of the lowest Lipschitz surface above the origin has an exponentially decaying tail. For sufficiently close to 1, the connected regions of ℤ−1 above which the surface has height 2 or more exhibit stretched-exponential tail behaviour....
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