Convergence of simple random walks on random discrete trees to brownian motion on the continuum random tree

David Croydon

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 6, page 987-1019
  • ISSN: 0246-0203

Abstract

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In this article it is shown that the brownian motion on the continuum random tree is the scaling limit of the simple random walks on any family of discrete n-vertex ordered graph trees whose search-depth functions converge to the brownian excursion as n→∞. We prove both a quenched version (for typical realisations of the trees) and an annealed version (averaged over all realisations of the trees) of our main result. The assumptions of the article cover the important example of simple random walks on the trees generated by the Galton–Watson branching process, conditioned on the total population size.

How to cite

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Croydon, David. "Convergence of simple random walks on random discrete trees to brownian motion on the continuum random tree." Annales de l'I.H.P. Probabilités et statistiques 44.6 (2008): 987-1019. <http://eudml.org/doc/78009>.

@article{Croydon2008,
abstract = {In this article it is shown that the brownian motion on the continuum random tree is the scaling limit of the simple random walks on any family of discrete n-vertex ordered graph trees whose search-depth functions converge to the brownian excursion as n→∞. We prove both a quenched version (for typical realisations of the trees) and an annealed version (averaged over all realisations of the trees) of our main result. The assumptions of the article cover the important example of simple random walks on the trees generated by the Galton–Watson branching process, conditioned on the total population size.},
author = {Croydon, David},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {continuum random tree; brownian motion; random graph tree; random walk; scaling limit; Brownian motion},
language = {eng},
number = {6},
pages = {987-1019},
publisher = {Gauthier-Villars},
title = {Convergence of simple random walks on random discrete trees to brownian motion on the continuum random tree},
url = {http://eudml.org/doc/78009},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Croydon, David
TI - Convergence of simple random walks on random discrete trees to brownian motion on the continuum random tree
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 6
SP - 987
EP - 1019
AB - In this article it is shown that the brownian motion on the continuum random tree is the scaling limit of the simple random walks on any family of discrete n-vertex ordered graph trees whose search-depth functions converge to the brownian excursion as n→∞. We prove both a quenched version (for typical realisations of the trees) and an annealed version (averaged over all realisations of the trees) of our main result. The assumptions of the article cover the important example of simple random walks on the trees generated by the Galton–Watson branching process, conditioned on the total population size.
LA - eng
KW - continuum random tree; brownian motion; random graph tree; random walk; scaling limit; Brownian motion
UR - http://eudml.org/doc/78009
ER -

References

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