2006/06/24 by Yuval Peres, Peres, Yuval, Ofer Zeitouni +1
Mathematics · Physics and Astronomy · #60F05 #60J80 #60K37 #82C41 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:60F05 #msc:60J80 #msc:60K37 #msc:82C41
paper · pdf · doi:10.48550/arxiv.math/0606625
34 pages, 4 figures
arxiv created 2006/06/24 · openalex publication_date 2006/06/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \cal 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 \cal 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 \v=limn |Xn|/n (where |Xn| is the distance of Xn from the root), with \v>0 for 0m 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]|/√(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.