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

TL;DR

This paper extends the asymptotic shape theorem for first-passage percolation to random environments formed by supercritical Bernoulli percolation, showing convergence to a deterministic shape and analyzing chemical distances.

Contribution

It generalizes the asymptotic shape result to percolation on random environments and establishes convergence to a deterministic shape independent of the environment.

Findings

Convergence of the renormalized wet points set to a deterministic shape.

Asymptotic shape theorem for chemical distance in supercritical Bernoulli percolation.

Proof of a flat edge result in the shape.

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.