Weak shape theorem in first passage percolation with infinite passage times

Raphaël Cerf (1, 2) , Marie Théret (3)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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]

Mots clés: First passage percolation; time constant; shape theorem.; shape theorem

Domaines: Probabilités [math.PR]

Raphaël Cerf, Marie Théret. Weak shape theorem in first passage percolation with infinite passage times. 2014. ⟨hal-00980479⟩