A Central Limit Theorem for biased random walks on Galton-Watson trees
| dc.creator | Peres, Yuval | |
| dc.creator | Zeitouni, Ofer | |
| dc.date | 2006-06-24 | |
| dc.date.accessioned | 2026-07-07T07:17:40Z | |
| dc.date.available | 2026-07-07T07:17:40Z | |
| dc.description | Let ${\cal T}$ be a rooted Galton-Watson tree with offspring distribution $\{p_k\}$ that has $p_0=0$, mean $m=\sum kp_k>1$ and exponential tails. Consider the $λ$-biased random walk $\{X_n\}_{n\geq 0}$ on ${\cal T}$; this is the nearest neighbor random walk which, when at a vertex $v$ with $d_v$ offspring, moves closer to the root with probability $λ/(λ+d_v)$, and moves to each of the offspring with probability $1/(λ+d_v)$. It is known that this walk has an a.s. constant speed $\v=\lim_n |X_n|/n$ (where $|X_n|$ is the distance of $X_n$ from the root), with $\v>0$ for $ 0<λ<m$ and $\v=0$ for $λ\ge m$. For all $λ\le m$, we prove a quenched CLT for $|X_n|-n\v$. (For $λ>m$ the walk is positive recurrent, and there is no CLT.) The most interesting case by far is $λ=m$, where the CLT has the following form: for almost every ${\cal T}$, the ratio $|X_{[nt]}|/\sqrt{n}$ converges in law as $n \to \infty$ to a deterministic multiple of the absolute value of a Brownian motion. Our approach to this case is based on an explicit description of an invariant measure for the walk from the point of view of the particle (previously, such a measure was explicitly known only for $λ=1$) and the construction of appropriate harmonic coordinates. | |
| dc.description | 34 pages, 4 figures | |
| dc.identifier | https://arxiv.org/abs/math/0606625 | |
| dc.identifier | http://arxiv.org/abs/math/0606625 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/114020 | |
| dc.subject | Probability | |
| dc.subject | 60K37; 60F05; 60J80; 82C41 | |
| dc.title | A Central Limit Theorem for biased random walks on Galton-Watson trees | |
| dc.type | text |