Abstract
Let $\mathcal{T}$ 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 $(X_t)_{t\ge0}$ on $\mathcal{T}$ 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 $\lambda\ge\rho$ and transient for $0\le\lambda<\rho$, 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 $\mathcal{T}$, the process $|X_{\lfloor nt \rfloor}|/\sqrt{n}$ converges in law as $n\to\infty$ 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.
Highlights
Let T denote an infinite tree with root o
The λ-biased random walk on T, hereafter denoted RWλ(T), is the Markov chain (Xt)t≥0 with X0 = o such that given Xt = v with offspring number dv and v = o, Xt+1 equals the parent of v with probability λ/(λ + dv), and is uniformly distributed among the offspring of v otherwise
Subject to no leaves and finite exponential moments for the offspring distribution, a quenched CLT for RWλ (λ ≤ ρ) on single-type Galton–Watson trees was shown by Peres–Zeitouni [32], and extended to the setting of random walk with random environment (RWRE) by Faraud [13]
Summary
Let T denote an infinite tree with root o. For supercritical Galton–Watson trees without leaves, if ρ denotes the mean offspring number, RWλ is a.s. recurrent if and only if λ ≥ ρ Galton–Watson tree, the processes (|X nt |/ n)t≥0 converge in law to the absolute value of a (deterministically) scaled Brownian motion Their proof is based on the construction of harmonic coordinates and an explicit description of a reversing probability measure IGWR for RWρ “from the point of view of the particle.”. As in [32], the key ingredient in our proof is the construction of an explicit reversing (probability) measure IMGWR for RWλ from the point of view of the particle, generalizing IGWR to the multi-type setting, for λ at the critical value on the boundary between transience and recurrence. Note that (H1) and ρ > 1 together imply MGWa(X) < 1 for all a ∈ Q
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