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

Abstract : 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 $\lambda\in (0,m)$, 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 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 $\lambda$.
Type de document :
Pré-publication, Document de travail
2017
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https://hal.archives-ouvertes.fr/hal-01557744
Contributeur : Shen Lin <>
Soumis le : lundi 31 juillet 2017 - 18:22:04
Dernière modification le : jeudi 3 août 2017 - 01:08:23

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  • HAL Id : hal-01557744, version 2
  • ARXIV : 1707.01811

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INSMI | PMA | UPMC | USPC

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Shen Lin. Harmonic measure for biased random walk in a supercritical Galton-Watson tree . 2017. <hal-01557744v2>

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