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

Shen Lin ()
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Shen Lin: Université Pierre et Marie Curie

Journal of Theoretical Probability, 2018, vol. 31, issue 3, 1469-1511

Abstract: 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 $$\alpha \in (1,2]$$ α ∈ ( 1 , 2 ] . Let $$\mu _n$$ μ 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 $$\mu _n$$ μ n carried by a random vertex uniformly chosen from height n is approximately equal to $$n^{-\lambda _\alpha }$$ n - λ α , where the constant $$\lambda _\alpha >\frac{1}{\alpha -1}$$ λ α > 1 α - 1 depends only on the index $$\alpha $$ α . In the analogous continuous model, this constant $$\lambda _\alpha $$ λ α turns out to be the typical local dimension of the continuous harmonic measure. Using an explicit formula for $$\lambda _\alpha $$ λ α , we are able to show that $$\lambda _\alpha $$ λ α decreases with respect to $$\alpha \in (1,2]$$ α ∈ ( 1 , 2 ] , and it goes to infinity at the same speed as $$(\alpha -1)^{-2}$$ ( α - 1 ) - 2 when $$\alpha $$ α approaches 1.

Keywords: Size-biased Galton–Watson tree; Reduced tree; Harmonic measure; Uniform measure; Simple random walk and Brownian motion on trees; 60J80; 60G50; 60K37 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10959-017-0752-6

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