A Central Limit Theorem for biased random walks on Galton-Watson trees

Abstract

Let $\mathcal{T}$ be a rooted Galton-Watson tree with offspring distribution $\{p_k\}$ that has $p_0=0$, mean $m=\sum k{p}_k>1$ and exponential tails. Consider the $\lambda$-biased random walk $\{X_n{\}}_{n\ge 0}$ on $\mathcal{T}$; this is the nearest neighbor random walk which, when at a vertex $v$ with $d_v$ offspring, moves closer to the root with probability $\lambda /(\lambda +d_v)$, and moves to each of the offspring with probability $1/(\lambda +d_v)$. It is known that this walk has an a.s. constant speed $\check{=}{\lim }_n|X_n|/n$ (where $|X_n|$ is the distance of $X_n$ from the root), with $\check{>}0$ for $0<\lambda <m$ and $\check{=}0$ for $\lambda \ge m$. For all $\lambda \le m$, we prove a quenched CLT for $|X_n|-n\v$. (For $\lambda >m$ the walk is positive recurrent, and there is no CLT.) The most interesting case by far is $\lambda =m$, where the CLT has the following form: for almost every $\mathcal{T}$, 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 $\lambda =1$) and the construction of appropriate harmonic coordinates.

34 pages, 4 figures