# The number of open paths in oriented percolation

Abstract : We study the number $N_n$ of open paths of length $n$ in supercritical oriented percolation on $\Zd \times \N$, with $d \ge 1$. We prove that on the percolation event $\{\inf N_n>0\}$, $N_n^{1/n}$ almost surely converges to a positive deterministic constant. We also study the existence of directional limits. The proof relies on the introduction of adapted sequences of regenerating times, on subadditive arguments and on the properties of the coupled zone in supercritical oriented percolation.
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https://hal.archives-ouvertes.fr/hal-00916083
Contributor : Olivier Garet Connect in order to contact the contributor
Submitted on : Wednesday, March 4, 2015 - 10:06:23 AM
Last modification on : Saturday, October 16, 2021 - 11:18:03 AM
Long-term archiving on: : Friday, June 5, 2015 - 10:25:15 AM

### Files

compte-chemin-preprint-v3.pdf
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### Identifiers

• HAL Id : hal-00916083, version 3
• ARXIV : 1312.2571

### Citation

Olivier Garet, Jean-Baptiste Gouéré, Régine Marchand. The number of open paths in oriented percolation. 2013. ⟨hal-00916083v3⟩

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