Asymptotic shape for the chemical distance and first-passage percolation in random environment

Olivier Garet; Regine Marchand

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arXiv · math/0304144

Asymptotic shape for the chemical distance and first-passage percolation in random environment

Olivier Garet·Regine Marchand

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Abstract

The aim of this paper is to generalize the well-known asymptotic shape result for first-passage percolation on $\Zd$ 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 points to a deterministic shape that does not depend on the random environment. As a special case of the previous result, we obtain an asymptotic shape theorem for the chemical distance in supercritical Bernoulli percolation. We also prove a flat edge result. Some various examples are also given.

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Olivier Garet, Regine Marchand. 2003-04-11. Asymptotic shape for the chemical distance and first-passage percolation in random environment. https://arxiv.org/abs/math/0304144

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