The Annals of Probability

The harmonic measure of balls in random trees

Nicolas Curien and Jean-François Le Gall

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We study properties of the harmonic measure of balls in typical large discrete trees. For a ball of radius $n$ centered at the root, we prove that, although the size of the boundary is of order $n$, most of the harmonic measure is supported on a boundary set of size approximately equal to $n^{\beta}$, where $\beta\approx0.78$ is a universal constant. To derive such results, we interpret harmonic measure as the exit distribution of the ball by simple random walk on the tree, and we first deal with the case of critical Galton–Watson trees conditioned to have height greater than $n$. An important ingredient of our approach is the analogous continuous model (related to Aldous’ continuum random tree), where the dimension of harmonic measure of a level set of the tree is equal to $\beta$, whereas the dimension of the level set itself is equal to $1$. The constant $\beta$ is expressed in terms of the asymptotic distribution of the conductance of large critical Galton–Watson trees.

Article information

Ann. Probab. Volume 45, Number 1 (2017), 147-209.

Received: March 2014
Revised: July 2015
First available in Project Euclid: 26 January 2017

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Zentralblatt MATH identifier

Primary: 05C81: Random walks on graphs 31C05: Harmonic, subharmonic, superharmonic functions 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]
Secondary: 05C80: Random graphs [See also 60B20] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Harmonic measure Brownian motion random walk random tree Galton–Watson tree Hausdorff dimension conductance


Curien, Nicolas; Le Gall, Jean-François. The harmonic measure of balls in random trees. Ann. Probab. 45 (2017), no. 1, 147--209. doi:10.1214/15-AOP1050.

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