Mathematics – Probability
Scientific paper
2005-09-22
Mathematics
Probability
42 pages; 1 figure; 2004
Scientific paper
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 $a_i$. Here $\aa =(a_1, ..., a_b)$ stands for some non-degenerated fixed set of weights. We consider the range $\{W^{\ee}_n ; n\geq 0 \}$ that is a subtree of $\U_b $. It corresponds to a unique random rooted ordered tree that we denote by $\tau_{\epsilon}$. We rescale the edges of $\tau_{\epsilon}$ by a factor $\ee $ and we let $\ee$ go to 0: we prove that correlations due to frequent backtracking of the random walk only give rise to a deterministic phenomenon taken into account by a positive factor $\gamma (\aa)$. More precisely, we prove that $\tau_{\epsilon}$ converges to a continuum random tree encoded by two independent Brownian motions with drift conditioned to stay positive and scaled in time by $\gamma (\aa)$. We actually state the result in the more general case of a random walk on a tree with an infinite number of branches at each node ($b=\infty$) and for a general set of weights $\aa =(a_n, n\geq 0)$.
No associations
LandOfFree
Continuum tree limit for the range of random walks on regular trees does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Continuum tree limit for the range of random walks on regular trees, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Continuum tree limit for the range of random walks on regular trees will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-503761