Weak shape theorem in first passage percolation with infinite passage times

Raphaël Cerf; Marie Théret

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arXiv · 1404.4539

Weak shape theorem in first passage percolation with infinite passage times

Raphaël Cerf·Marie Théret

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Abstract

We consider the model of i.i.d. first passage percolation on $\mathbb{Z}^d$ : we associate with each edge $e$ of the graph a passage time $t(e)$ taking values in $[0,+\infty]$, such that $\mathbb{P}[t(e)<+\infty] >p_c(d)$. Equivalently, we consider a standard (finite) i.i.d. first passage percolation model on a super-critical Bernoulli percolation performed independently. We prove a weak shape theorem without any moment assumption. We also prove that the corresponding time constant is positive if and only if $\mathbb{P}[t(e)=0]<p_c(d)$.

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Raphaël Cerf, Marie Théret. 2014-11-20. Weak shape theorem in first passage percolation with infinite passage times. https://arxiv.org/abs/1404.4539

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