Electronic Journal of Probability

The harmonic measure of balls in critical Galton-Watson trees with infinite variance offspring distribution

Shen Lin

Full-text: Open access

Abstract

We study properties of the harmonic measure of balls in large critical Galton-Watson trees whose offspring distribution is in the domain of attraction of a stable distribution with index $\alpha\in (1,2]$. Here the harmonic measure refers to the hitting distribution of height $n$ by simple random walk on the critical Galton-Watson tree conditioned on non-extinction at generation $n$. For a ball of radius $n$ centered at the root, we prove that, although the size of the boundary is roughly of order $n^{\frac{1}{\alpha-1}}$, most of the harmonic measure is supported on a boundary subset of size approximately equal to $n^{\beta_{\alpha}}$, where the constant $\beta_{\alpha}\in (0,\frac{1}{\alpha-1})$ depends only on the index $\alpha$. Using an explicit expression of $\beta_{\alpha}$, we are able to show the uniform boundedness of $(\beta_{\alpha}, 1<\alpha\leq 2)$. These are generalizations of results in a recent paper of Curien and Le Gall.

Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 98, 35 pp.

Dates
Accepted: 20 October 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065740

Digital Object Identifier
doi:10.1214/EJP.v19-3498

Mathematical Reviews number (MathSciNet)
MR3272331

Zentralblatt MATH identifier
1325.60138

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

Keywords
critical Galton-Watson tree harmonic measure Hausdorff dimension invariant measure simple random walk and Brownian motion on trees

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Lin, Shen. The harmonic measure of balls in critical Galton-Watson trees with infinite variance offspring distribution. Electron. J. Probab. 19 (2014), paper no. 98, 35 pp. doi:10.1214/EJP.v19-3498. https://projecteuclid.org/euclid.ejp/1465065740


Export citation

References

  • E. Aïdékon: Speed of the biased random walk on a Galton–Watson tree. Probab. Theory Related Fields 159 (2014), no. 3-4, 597–617.
  • K. B. Athreya, P. E. Ney: Branching processes. Die Grundlehren der mathematischen Wissenschaften, Band 196. Springer-Verlag, New York-Heidelberg, 1972. xi+287 pp.
  • G. Ben Arous, A. Fribergh, and V. Sidoravicius: A proof of the Lyons-Pemantle-Peres monotonicity conjecture for high biases, to appear in Communications in Pure and Applied Mathematics. arXiv:1111.5865
  • N. H. Bingham, C. M. Goldie, and J. L. Teugels: Regular variation. Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge, 1987. xx+491 pp. ISBN: 0-521-30787-2.
  • N. Curien, J.-F. Le Gall: The harmonic measure of balls in random trees, preprint 2013, arxiv:1304.7190
  • T. Duquesne, J.-F. Le Gall: Random trees, Lévy processes and spatial branching processes. Astérisque No. 281 (2002), vi+147 pp.
  • T. Duquesne, J.-F. Le Gall: The Hausdorff measure of stable trees, Alea Lat. Am. J. Probab. Math. Stat., 1 (2006), 393-415.
  • W. Feller: An introduction to probability theory and its applications. Vol. II. Second edition John Wiley & Sons, Inc., New York-London-Sydney 1971 xxiv+669 pp.
  • K. FleÄ­shmann, V. A. Vatutin, and V. Wachtel: Critical Galton-Watson branching processes: the maximum of the total number of particles within a large window. (Russian) Teor. Veroyatn. Primen. 52 (2007), no. 3, 419–445; translation in Theory Probab. Appl. 52 (2008), no. 3, 470–492
  • J.-F. Le Gall: Random trees and applications. Probab. Surv. 2 (2005), 245–311.
  • S. Lin: Tree-indexed random walk and random walk on trees, PhD Thesis, Université Paris-Sud XI, 2014.
  • R. Lyons, R. Pemantle, and Y. Peres: Ergodic theory on Galton-Watson trees: speed of random walk and dimension of harmonic measure. Ergodic Theory Dynam. Systems 15 (1995), no. 3, 593–619.
  • R. Lyons, R. Pemantle, and Y. Peres: Biased random walks on Galton-Watson trees, Probab. Theory Related Fields, 106 (1996), 249–264.
  • R. Lyons and Y. Peres: Probability on Trees and Networks, Book in preparation, Current version available at http://mypage.iu.edu/~rdlyons/
  • B. Mehrdad, S. Sen, and L.-J. Zhu: The Speed of a Biased Walk on a Galton-Watson Tree without Leaves is Monotonic with Respect to Progeny Distributions for High Values of Bias, to appear in Annales de l'Institut Henri Poincaré. arxiv:1212.3004
  • R. S. Slack: A branching process with mean one and possibly infinite variance. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 9 1968 139–145.
  • V. A. Vatutin: Limit theorems for critical multitype Markov branching processes with infinite second moments. (Russian) Mat. Sb. (N.S.) 103(145) (1977), no. 2, 253–264, 319.
  • A. L. Yakymiv: Reduced branching processes. (Russian) Teor. Veroyatnost. i Primenen. 25 (1980), no. 3, 593–596; translation in Theory Probab. Appl. 25 (1980), no. 3, 584-588.