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Journal of Theoretical Probability

, Volume 31, Issue 3, pp 1469–1511 | Cite as

Typical Behavior of the Harmonic Measure in Critical Galton–Watson Trees with Infinite Variance Offspring Distribution

  • Shen Lin
Article
  • 56 Downloads

Abstract

We study the typical behavior of the harmonic measure in large critical Galton–Watson trees whose offspring distribution is in the domain of attraction of a stable distribution with index \(\alpha \in (1,2]\). Let \(\mu _n\) denote the hitting distribution of height n by simple random walk on the critical Galton–Watson tree conditioned on non-extinction at generation n. We extend the results of Lin (Typical behavior of the harmonic measure in critical Galton–Watson trees, arXiv:1502.05584, 2015) to prove that, with high probability, the mass of the harmonic measure \(\mu _n\) carried by a random vertex uniformly chosen from height n is approximately equal to \(n^{-\lambda _\alpha }\), where the constant \(\lambda _\alpha >\frac{1}{\alpha -1}\) depends only on the index \(\alpha \). In the analogous continuous model, this constant \(\lambda _\alpha \) turns out to be the typical local dimension of the continuous harmonic measure. Using an explicit formula for \(\lambda _\alpha \), we are able to show that \(\lambda _\alpha \) decreases with respect to \(\alpha \in (1,2]\), and it goes to infinity at the same speed as \((\alpha -1)^{-2}\) when \(\alpha \) approaches 1.

Keywords

Size-biased Galton–Watson tree Reduced tree Harmonic measure Uniform measure Simple random walk and Brownian motion on trees 

Mathematics Subject Classification (2010)

60J80 60G50 60K37 

Notes

Acknowledgements

The author is grateful to an anonymous referee for many comments that greatly improved the paper.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.LPMAUniversité Pierre et Marie CurieParisFrance

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