@article{EJP3498,
author = {Shen Lin},
title = {The harmonic measure of balls in critical Galton-Watson trees with infinite variance offspring distribution},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {19},
year = {2014},
keywords = {critical Galton-Watson tree; harmonic measure; Hausdorff dimension; invariant measure; simple random walk and Brownian motion on trees},
abstract = {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.},
pages = {no. 97, 1-35},
issn = {1083-6489},
doi = {10.1214/EJP.v19-3498},
url = {http://ejp.ejpecp.org/article/view/3498}}