## The Annals of Probability

- Ann. Probab.
- Volume 45, Number 6A (2017), 4071-4100.

### The number of open paths in oriented percolation

Olivier Garet, Jean-Baptiste Gouéré, and Régine Marchand

#### Abstract

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.

#### Article information

**Source**

Ann. Probab., Volume 45, Number 6A (2017), 4071-4100.

**Dates**

Received: November 2015

Revised: September 2016

First available in Project Euclid: 27 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1511773672

**Digital Object Identifier**

doi:10.1214/16-AOP1158

**Mathematical Reviews number (MathSciNet)**

MR3729623

**Zentralblatt MATH identifier**

06838115

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 82B43: Percolation [See also 60K35]

**Keywords**

Subadditive ergodic theorem oriented percolation

#### Citation

Garet, Olivier; Gouéré, Jean-Baptiste; Marchand, Régine. The number of open paths in oriented percolation. Ann. Probab. 45 (2017), no. 6A, 4071--4100. doi:10.1214/16-AOP1158. https://projecteuclid.org/euclid.aop/1511773672