DUQUESNE, Thomas. - Modélisation Stochastique et Statistique, Université Paris-Sud, Bât. 425, 91405 Orsay cedex

Abstract :

Let $b$ be an integer greater than $1$ and let $W^{\ee}=(W^{\ee}_n; n\geq 0)$ be a random walk on the $b$-ary rooted tree $\U_b$ starting at the root, going up (resp. down) with probability $1/2+\epsilon$ (resp. $1/2 -\epsilon$) , $\epsilon \in (0, 1/2)$, and choosing direction $i\in \{ 1, ... , b}$ when going up with probability ai. Here $\aa =(a1 , ... , ab)$ is some non-degenerated fixed set of weights. We consider the range $\{ W^{\ee}_n ; n\geq 0 \}$ which is a subtree of $\U_b $. It corresponds to a unique random rooted ordered tree $\tau_{\epsilon}$ that we study when $\ee$ goes to 0: We rescale the edges by a factor $\ee $ and we prove that the resulting graph converges to a continuum random tree coded by a deterministic constant $\gamma (\aa)$ which takes into account the long range correlations of $\tau_{\ee}$ and by two independent Brownian motions with drift conditioned to stay positive.

Article : Fichier Postscript
Contact : Thomas.Duquesne@math.u-psud.fr