Paper detail

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

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 $α\in (1,2]$. 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 $n^{\frac{1}{α-1}}$, most of the harmonic measure is supported on a boundary subset of size approximately equal to $n^{β_α}$, where the constant $β_α\in (0,\frac{1}{α-1})$ depends only on the index $α$. Using an explicit expression of $β_α$, we are able to show the uniform boundedness of $(β_α, 1<α\leq 2)$. These are generalizations of results in a recent paper of Curien and Le Gall (arXiv: 1304.7190).