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

Croydon, David

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 6, pp. 987-1019.

Dans cet article, nous démontrons qu’un mouvement brownien sur un arbre aléatoire continu est en fait la limite rééchelonnée d’un certain type de marches aléatoires simples; ces marches aléatoires simples évoluent sur n’importe quelle famille de graphes d’arbres discrets ordonnés de $n$ sommets, dont les fonctions de recherche en profondeur convergent vers une excursion brownienne lorsque $n\to \infty$. Nous prouvons deux versions de notre résultat principal: une première conditionnelle sur les réalisations typiques des arbres, ainsi qu’une seconde où l’on prend la moyenne sur toutes les réalisations des arbres. Les hypothèses de cet article couvrent l’exemple important d’une marche aléatoire simple sur les arbres générés par le processus de branchement de Galton-Watson, étant donné la taille de la population totale.

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 $n\to \infty$. 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.

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