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

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 $({\beta }_{\alpha },1<\alpha \le 2)$. These are generalizations of results in a recent paper of Curien and Le Gall (arXiv: 1304.7190).

46 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:1304.7190 by other authors