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Harmonic measure for biased random walk in a supercritical Galton-Watson tree. (arXiv:1707.01811v1 [math.PR])
来源于:arXiv
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 查看全文>>