Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Barrier estimates for a critical Galton–Watson process and the cover time of the binary tree

David Belius, Jay Rosen, and Ofer Zeitouni

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Abstract

For the critical Galton–Watson process with geometric offspring distributions we provide sharp barrier estimates for barriers which are (small) perturbations of linear barriers. These are useful in analyzing the cover time of finite graphs in the critical regime by random walk, and the Brownian cover times of compact two-dimensional manifolds. As an application of the barrier estimates, we prove that if $C_{L}$ denotes the cover time of the binary tree of depth $L$ by simple walk, then $\sqrt{C_{L}/2^{L+1}}-\sqrt{2\log2}L+\log L/\sqrt{2\log2}$ is tight. The latter improves results of Aldous (J. Math. Anal. Appl. 157 (1991) 271–283), Bramson and Zeitouni (Ann. Probab. 37 (2009) 615–653) and Ding and Zeitouni (Stochastic Process. Appl. 122 (2012) 2117–2133). In a subsequent article we use these barrier estimates to prove tightness of the Brownian cover time for compact two-dimensional manifolds.

Résumé

Pour le processus critique de Galton–Watson avec loi de reproduction géométrique de la progéniture, nous fournissons des estimations fines de barrière pour des obstacles qui sont des (petites) perturbations de barrières linéaires. Les estimations sont utiles pour analyser le temps de recouvrement, par une marche aleatoire, de graphes finis dans le régime critique, et les temps de recouvrement brownien de variétés bidemensionelles compactes. Comme application des estimations de barrière, nous prouvons que si $C_{L}$ dénote le temps de recouvrement de l’arbre binaire de profondeur $L$ par une marche aleatoire simple, la suite $\sqrt{C_{L}/2^{L+1}}-\sqrt{2\log2}L+\log L/\sqrt{2\log2}$ est tendue. Ce dernier resultat améliore les résultats d’Aldous (J. Math. Anal. Appl. 157 (1991) 271–283), Bramson et Zeitouni (Ann. Probab. 37 (2009) 615–653) et Ding et Zeitouni (Stochastic Process. Appl. 122 (2012) 2117–2133). Dans un article compagnon, nous utilisons ces estimations de barrière pour prouver la tension du temps de recouvrement brownien pour des variétés riemanniennes compactes en deux dimensions.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 1 (2019), 127-154.

Dates
Received: 13 February 2017
Revised: 13 October 2017
Accepted: 11 December 2017
First available in Project Euclid: 18 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1547802397

Digital Object Identifier
doi:10.1214/17-AIHP878

Mathematical Reviews number (MathSciNet)
MR3901643

Zentralblatt MATH identifier
07039767

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J85: Applications of branching processes [See also 92Dxx] 60G50: Sums of independent random variables; random walks

Keywords
Galton–Watson process Cover time Binary tree Barrier estimates

Citation

Belius, David; Rosen, Jay; Zeitouni, Ofer. Barrier estimates for a critical Galton–Watson process and the cover time of the binary tree. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 1, 127--154. doi:10.1214/17-AIHP878. https://projecteuclid.org/euclid.aihp/1547802397


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