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- W2289353885 abstract "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 $$alpha in (1,2]$$ . Let $$mu _n$$ 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 Lin (Typical behavior of the harmonic measure in critical Galton–Watson trees, arXiv:1502.05584 , 2015) to prove that, with high probability, the mass of the harmonic measure $$mu _n$$ carried by a random vertex uniformly chosen from height n is approximately equal to $$n^{-lambda _alpha }$$ , where the constant $$lambda _alpha >frac{1}{alpha -1}$$ depends only on the index $$alpha $$ . In the analogous continuous model, this constant $$lambda _alpha $$ turns out to be the typical local dimension of the continuous harmonic measure. Using an explicit formula for $$lambda _alpha $$ , we are able to show that $$lambda _alpha $$ decreases with respect to $$alpha in (1,2]$$ , and it goes to infinity at the same speed as $$(alpha -1)^{-2}$$ when $$alpha $$ approaches 1." @default.
- W2289353885 created "2016-06-24" @default.
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- W2289353885 date "2017-03-16" @default.
- W2289353885 modified "2023-10-09" @default.
- W2289353885 title "Typical Behavior of the Harmonic Measure in Critical Galton–Watson Trees with Infinite Variance Offspring Distribution" @default.
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- W2289353885 doi "https://doi.org/10.1007/s10959-017-0752-6" @default.
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