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January 2017 The harmonic measure of balls in random trees
Nicolas Curien, Jean-François Le Gall
Ann. Probab. 45(1): 147-209 (January 2017). DOI: 10.1214/15-AOP1050

Abstract

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.

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Nicolas Curien. Jean-François Le Gall. "The harmonic measure of balls in random trees." Ann. Probab. 45 (1) 147 - 209, January 2017. https://doi.org/10.1214/15-AOP1050

Information

Received: 1 March 2014; Revised: 1 July 2015; Published: January 2017
First available in Project Euclid: 26 January 2017

zbMATH: 1358.05084
MathSciNet: MR3601648
Digital Object Identifier: 10.1214/15-AOP1050

Subjects:
Primary: 05C81 , 31C05 , 60J45
Secondary: 05C80 , 60J80

Keywords: Brownian motion , conductance , Galton–Watson tree , harmonic measure , Hausdorff dimension , Random tree , Random walk

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 1 • January 2017
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