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Scaling limits for simple random walks on random ordered graph trees

Part of: Markov processes Special processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

D. A. Croydon
D. A. Croydon*
Affiliation:
University of Warwick
*
∗ Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email address: [email protected]
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Abstract

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Consider a family of random ordered graph trees (Tn)n≥1, where Tn has n vertices. It has previously been established that if the associated search-depth processes converge to the normalised Brownian excursion when rescaled appropriately as n → ∞, then the simple random walks on the graph trees have the Brownian motion on the Brownian continuum random tree as their scaling limit. Here, this result is extended to demonstrate the existence of a diffusion scaling limit whenever the volume measure on the limiting real tree is nonatomic, supported on the leaves of the limiting tree, and satisfies a polynomial lower bound for the volume of balls. Furthermore, as an application of this generalisation, it is established that the simple random walks on a family of Galton-Watson trees with a critical infinite variance offspring distribution, conditioned on the total number of offspring, can be rescaled to converge to the Brownian motion on a related α-stable tree.

Keywords

Branching processrandom graph treerandom walkscaling limitα-stable tree

MSC classification

Primary: 60K37: Processes in random environments
Secondary: 60G99: None of the above, but in this section 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 2 , June 2010 , pp. 528 - 558
Copyright
Copyright © Applied Probability Trust 2010 

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