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November 2017 The number of open paths in oriented percolation
Olivier Garet, Jean-Baptiste Gouéré, Régine Marchand
Ann. Probab. 45(6A): 4071-4100 (November 2017). DOI: 10.1214/16-AOP1158


We study the number $N_{n}$ of open paths of length $n$ in supercritical oriented percolation on $\mathbb{Z}^{d}\times\mathbb{N}$, with $d\ge1$, and we prove the existence of the connective constant for the supercritical oriented percolation cluster: on the percolation event $\{\inf N_{n}>0\}$, $N_{n}^{1/n}$ almost surely converges to a positive deterministic constant.

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. This global convergence result can be deepened to give directional limits and can be extended to more general random linear recursion equations known as linear stochastic evolutions.


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Olivier Garet. Jean-Baptiste Gouéré. Régine Marchand. "The number of open paths in oriented percolation." Ann. Probab. 45 (6A) 4071 - 4100, November 2017.


Received: 1 November 2015; Revised: 1 September 2016; Published: November 2017
First available in Project Euclid: 27 November 2017

zbMATH: 06838115
MathSciNet: MR3729623
Digital Object Identifier: 10.1214/16-AOP1158

Primary: 60K35
Secondary: 82B43

Keywords: Oriented percolation , subadditive ergodic theorem

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 6A • November 2017
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