2016/03/03 by Shen Lin, Lin, Shen
Mathematics · Physics and Astronomy · #60G50 #60J80 #60K37 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1603.01200
openalex publication_date 2016/03/03 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
We study the typical behavior of the harmonic measure in large critical Galton-Watson trees whose offspring distribution is in the domain of attraction of a stable distribution with index α∈ (1,2]. Let μn denote the hitting distribution of height n by simple random walk on the critical Galton-Watson tree conditioned on non-extinction at generation n. We extend the results of arxiv:1502.05584 to prove that, with high probability, the mass of the harmonic measure μn carried by a random vertex uniformly chosen from height n is approximately equal to n-λα, where the constant λα>(1)/(α-1) depends only on the index α. In the analogous continuous model, this constant λα turns out to be the typical local dimension of the continuous harmonic measure. Using an explicit formula for λα, we are able to show that λα decreases with respect to α∈(1,2], and it goes to infinity at the same speed as (α-1)-2 when α approaches 1.