Strict convexity of the limit shape in first-passage percolation.

Lalley, Steven P.

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The aim of this paper is to extend the well-known asymptotic shape result for first-passage percolation on $ℤ^{d}$ to first-passage percolation on a random environment given by the infinite cluster of a supercritical Bernoulli percolation model. We prove the convergence of the renormalized set of wet vertices to a deterministic shape that does not depend on the realization of the infinite cluster. As a special case of our result, we obtain an asymptotic shape theorem for the chemical distance...