The harmonic measure of balls in random trees
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Nicolas Curien [1] ; Jean-François Le Gall [1]
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[1] University of Paris-Sud
University of Paris-Sud
Arrondissement de Palaiseau, Francia
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 1, 2017, págs. 147-209
- Idioma: inglés
- DOI: 10.1214/15-AOP1050
- Enlaces
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Resumen
We study properties of the harmonic measure of balls in typical large discrete trees. For a ball of radius nn centered at the root, we prove that, although the size of the boundary is of order nn, most of the harmonic measure is supported on a boundary set of size approximately equal to nβnβ, where β≈0.78β≈0.78 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 nn. 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 ββ, whereas the dimension of the level set itself is equal to 11. The constant ββ is expressed in terms of the asymptotic distribution of the conductance of large critical Galton–Watson trees.
