Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Typical behavior of the harmonic measure in critical Galton–Watson trees

Shen Lin

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Abstract

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.

Résumé

Nous étudions le comportement typique de la mesure harmonique au bord des boules dans les grands arbres de Galton–Watson critiques, dont la loi de reproduction est de variance finie. On comprend par mesure harmonique la loi du premier point d’atteinte de la hauteur $n$ par une marche aléatoire simple sur un arbre de Galton–Watson critique conditionné à avoir une hauteur supérieure à $n$. Nous prouvons que, avec une grande probabilité, la masse de la mesure harmonique portée par un sommet choisi uniformément au hasard de la hauteur $n$ est approximativement égale à $n^{-\lambda}$, où la constante $\lambda>1$ ne dépend pas de la loi de reproduction. Cette constante universelle $\lambda$ est égale au moment d’ordre 1 de la distribution asymptotique de la conductance de l’arbre de Galton–Watson biaisé par la taille moins 1.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 2 (2017), 718-752.

Dates
Received: 19 February 2015
Revised: 20 November 2015
Accepted: 20 November 2015
First available in Project Euclid: 11 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1491897743

Digital Object Identifier
doi:10.1214/15-AIHP734

Mathematical Reviews number (MathSciNet)
MR3634272

Zentralblatt MATH identifier
1367.60104

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G50: Sums of independent random variables; random walks 60K37: Processes in random environments

Keywords
Size-biased Galton–Watson tree Harmonic measure Uniform measure Simple random walk and Brownian motion on trees

Citation

Lin, Shen. Typical behavior of the harmonic measure in critical Galton–Watson trees. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 2, 718--752. doi:10.1214/15-AIHP734. https://projecteuclid.org/euclid.aihp/1491897743


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