2014/05/07 by Shen Lin, Lin, Shen
Mathematics · #60G50 #60J80 #60K37 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60G50 #msc:60J80 #msc:60K37
paper · pdf · doi:10.48550/arxiv.1405.1583
46 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:1304.7190 by other authors
arxiv created 2014/05/07 · arxiv updated 2014/05/08
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 nβα, 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 (arXiv: 1304.7190).