Electronic Journal of Probability

The harmonic measure of balls in critical Galton-Watson trees with infinite variance offspring distribution

Shen Lin

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We study properties of the harmonic measure of balls in large critical Galton-Watson trees whose offspring distribution is in the domain of attraction of a stable distribution with index $\alpha\in (1,2]$. Here the harmonic measure refers to the hitting distribution of height $n$ by simple random walk on the critical Galton-Watson tree conditioned on non-extinction at generation $n$. For a ball of radius $n$ centered at the root, we prove that, although the size of the boundary is roughly of order $n^{\frac{1}{\alpha-1}}$, most of the harmonic measure is supported on a boundary subset of size approximately equal to $n^{\beta_{\alpha}}$, where the constant $\beta_{\alpha}\in (0,\frac{1}{\alpha-1})$ depends only on the index $\alpha$. Using an explicit expression of $\beta_{\alpha}$, we are able to show the uniform boundedness of $(\beta_{\alpha}, 1<\alpha\leq 2)$. These are generalizations of results in a recent paper of Curien and Le Gall.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 98, 35 pp.

Accepted: 20 October 2014
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G50: Sums of independent random variables; random walks 60K37: Processes in random environments

critical Galton-Watson tree harmonic measure Hausdorff dimension invariant measure simple random walk and Brownian motion on trees

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Lin, Shen. The harmonic measure of balls in critical Galton-Watson trees with infinite variance offspring distribution. Electron. J. Probab. 19 (2014), paper no. 98, 35 pp. doi:10.1214/EJP.v19-3498. https://projecteuclid.org/euclid.ejp/1465065740

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