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

Let $${\mathcal{T}}$$ be a rooted Galton–Watson tree with offspring distribution {pk} that has p0 = 0, mean m = ∑ kpk > 1 and exponential tails. Consider the λ-biased random walk {Xn}n ≥ 0 on $${\mathcal{T}}$$ ; this is the nearest neighbor random walk which, when at a vertex v with dv offspring, moves closer to the root with probability λ/(λ + dv), and moves to each of the offspring with probability 1/(λ + dv). It is known that this walk has an a.s. constant speed $${\tt v} = \lim_n |X_n|/n$$ (where |Xn| is the distance of Xn from the root), with $${\tt v} > 0$$ for 0 < λ < m and $${\tt v} = 0$$ for λ ≥ m. For all λ ≤ m, we prove a quenched CLT for $$|X_n| - n{\tt v}$$ . (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 $${\mathcal{T}}$$ , the ratio $$|X_{[nt]}|/\sqrt{n}$$ 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.

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