  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.
