Typical behavior of the harmonic measure in critical Galton-Watson trees

We study the typical behavior of the harmonic measure of balls in large critical Galton-Watson trees whose offspring distribution has finite variance. The harmonic measure considered here refers to the hitting distribution of height $n$ by simple random walk on a critical Galton-Watson tree conditioned to have height greater than $n$. We prove that, with high probability, the mass of the harmonic measure carried by a random vertex uniformly chosen from height $n$ is approximately equal to $n^{-λ}$, where the constant $λ>1$ does not depend on the offspring distribution. This universal constant $λ$ is equal to the first moment of the asymptotic distribution of the conductance of size-biased Galton-Watson trees minus 1.

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