The number of open paths in oriented percolation

• Olivier Garet ^[1] ; Jean-Baptiste Gouéré ^[2] ; Régine Marchand ^[1]
  1. [1] University of Lorraine

     University of Lorraine

     Arrondissement de Nancy, Francia

     
  2. [2] Université de Tours (France)
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 6, 1, 2017, págs. 4071-4100
• Idioma: inglés
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• Resumen

  • We study the number NnNn of open paths of length nn in supercritical oriented percolation on Zd×NZd×N, with d≥1d≥1, and we prove the existence of the connective constant for the supercritical oriented percolation cluster: on the percolation event {infNn>0}{infNn>0}, N1/nnNn1/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.