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 ∈ (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 1 α−1 , most of the harmonic measure is supported on a boundary subset of size approximately equal to nα , where the constant βα ∈ (0, 1 α−1 ) depends only on the index α. Using an explicit expression of βα, we are able to show the uniform boundedness of (βα, 1 < α ≤ 2). These are generalizations of results in a recent paper of Curien and Le Gall [5].