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(-lambda), where the constant lambda > 1 does not depend on the offspring distribution. This universal constant lambda is equal to the first moment of the asymptotic distribution of the conductance of size-biased Galton-Watson trees minus 1.