Central limit theorem for biased random walk on multi-type Galton-Watson trees

Details Central limit theorem for biased random walk on multi-type Galton-Watson trees Central limit theorem for biased random walk on multi-type Galton-Watson trees

2010-11-17

arxiv.org/abs/1011.4056v1

Mathematics

Probability

28 pages

Scientific paper

Let T be a rooted multi-type Galton-Watson (MGW) tree of finitely many types with at least one offspring at each vertex, and an offspring distribution with exponential tails. The lambda-biased random walk (X_t, t>=0) on T 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 to each of the offspring with probability 1/(lambda+d(v)). This walk is recurrent for lambda >= rho and transient for 0 <= lambda < rho, with rho the Perron-Frobenius eigenvalue for the (assumed) irreducible matrix of expected offspring numbers. We prove the following quenched CLT for the critical value lambda = rho: for almost every T, the process |X_{floor(nt)}|/sqrt{n} converges in law as n tends to infinity to a deterministic positive multiple of a reflected Brownian motion. Following the approach of Peres and Zeitouni (2008) for Galton-Watson trees, our proof is based on a new explicit description of a reversing measure for the walk from the point of view of the particle (Peres and Zeitouni provide this measure only for Galton-Watson trees), and the construction of appropriate harmonic coordinates. Finally, we extend our construction of the reversing measure to a biased random walk with random environment (RWRE) on MGW trees, again at a critical value of the bias. We compare this result against a transience-recurrence criterion for the RWRE generalizing a result of Faraud (2008) for Galton-Watson trees.

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