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

TL;DR

This paper establishes a central limit theorem for a biased random walk on multi-type Galton-Watson trees at a critical bias, using new harmonic coordinates and a reversing measure, extending previous results to more complex tree structures.

Contribution

It introduces a novel explicit reversing measure and harmonic coordinates for the biased walk on multi-type Galton-Watson trees, generalizing prior single-type results.

Findings

Proves quenched CLT for the critical bias case.

Constructs a new reversing measure for the walk.

Extends the analysis to biased random walks in random environments on MGW trees.

Abstract

Let T be a rooted supercritical multi-type Galton-Watson (MGW) tree with types coming from a finite alphabet, conditioned to non-extinction. 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. Subject to finite moments of order p>4 for the offspring distributions, we prove the following quenched CLT for lambda-biased random walk at 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 reflected Brownian motion rescaled by an explicit constant. This result was proved under some stronger assumptions by Peres-Zeitouni (2008) for single-type Galton-Watson trees. Following their approach, our proof is based on a new explicit description of a reversing measure for the walk from the point of view of the particle (generalizing the measure constructed in the single-type setting by Peres-Zeitouni), and the construction of appropriate harmonic coordinates. In carrying out this program we prove moment and conductance estimates for MGW trees, which may be of independent interest. In addition, 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 (2011) for Galton-Watson trees.