A central limit theorem for biased random walks on Galton-Watson trees

Abstract

Let T be a rooted Galton-Watson tree with offspring distribution {p k} that has p 0 = 0, mean m = kp k > 1 and exponential tails. Consider the λ-biased random walk {X n} n 0 on 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 (where |X n| is the distance of X n from the root), with > 0 for 0 < 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 T, the ratio converges in law as n → ∞ 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. © 2007 Springer-Verlag.