THE HARMONIC MEASURE OF BALLS IN RANDOM TREES

• 成果类型: Article
• 署名作者: Curien, Nicolas; Le Gall, Jean-Francois
• 署名单位: Universite Paris Saclay
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1050
• 发表日期: 2017
• 页码: 147-209
• 关键词: galton-watson trees; brownian-motion; random-walks; dimension; graphs

摘要

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 approximate to 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 Gallon 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 13, whereas the dimension of the level set itself is equal to 1. The constant 13 is expressed in terms of the asymptotic distribution of the conductance of large critical Galton-Watson trees.