# Weak shape theorem in first passage percolation with infinite passage times

* Corresponding author
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]
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https://hal.archives-ouvertes.fr/hal-00980479
Contributor : Marie Théret <>
Submitted on : Friday, April 18, 2014 - 9:58:34 AM
Last modification on : Friday, March 27, 2020 - 3:31:40 AM

### Identifiers

• HAL Id : hal-00980479, version 1
• ARXIV : 1404.4539

### Citation

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

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