The harmonic measure of balls in critical Galton-Watson trees with infinite variance offspring distribution. (English) Zbl 1325.60138

Lin, Shen

Electron. J. Probab. 19, Paper No. 98, 35 p. (2014).

Summary: 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 N. Curien and J.-F. Le Gall [“The harmonic measure of balls in random trees”, Preprint, arXiv:1304.7190].

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60G50	Sums of independent random variables; random walks
60J65	Brownian motion
60K37	Processes in random environments

Keywords:

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