Harmonic measure for biased random walk in a supercritical Galton-Watson tree

Shen Lin

Published 2017-07-06Version 1

We consider random walks $\lambda$-biased towards the root on a Galton-Watson tree, whose offspring distribution $(p_k)_{k\geq 1}$ is non-degenerate and has finite mean $m>1$. In the transient regime $0<\lambda <m$, the loop-erased trajectory of the biased random walk defines the $\lambda$-harmonic ray, whose law is the $\lambda$-harmonic measure on the boundary of the Galton-Watson tree. We answer a question of Lyons, Pemantle and Peres \cite{LPP97} by showing that the $\lambda$-harmonic measure has a.s. strictly larger Hausdorff dimension than that of the visibility measure. We also prove that the average number of children of the vertices visited by the $\lambda$-harmonic ray is a.s. bounded below by $m$ and bounded above by $m^{-1}\sum k^2 p_k$. Moreover, the average number of children along the $\lambda$-harmonic ray is a.s. strictly larger than the average number of children along the $\lambda$-biased random walk trajectory. We observe that the latter is not monotone in the bias parameter $\lambda$.