The range of tree-indexed random walk in low dimensions

• Le Gall, Jean-François ^[1] ; Lin, Shen ^[2]
  1. [1] University of Paris-Sud

     University of Paris-Sud

     Arrondissement de Palaiseau, Francia

     
  2. [2] École Normale Supérieure

     École Normale Supérieure

     Francia

     
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 43, Nº. 5, 2015, págs. 2701-2728
• Idioma: inglés
• DOI: 10.1214/14-AOP947
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• Resumen

  • We study the range Rn of a random walk on the d-dimensional lattice Zd indexed by a random tree with n vertices. Under the assumption that the random walk is centered and has finite fourth moments, we prove in dimension d≤3 that n−d/4Rn converges in distribution to the Lebesgue measure of the support of the integrated super-Brownian excursion (ISE). An auxiliary result shows that the suitably rescaled local times of the tree-indexed random walk converge in distribution to the density process of ISE. We obtain similar results for the range of critical branching random walk in Zd, d≤3. As an intermediate estimate, we get exact asymptotics for the probability that a critical branching random walk starting with a single particle at the origin hits a distant point. The results of the present article complement those derived in higher dimensions in our earlier work.