Weak shape theorem in first passage percolation with infinite passage times

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 $\mathbb{P}[t(e)=0]<p_c(d)$.

35 pages, 4 figures; minor changes between v1 and v2, some references have been added