Advances in Applied Probability

Scaling limits for simple random walks on random ordered graph trees

D. A. Croydon

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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.

Article information

Adv. in Appl. Probab. Volume 42, Number 2 (2010), 528-558.

First available in Project Euclid: 28 May 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments
Secondary: 60G99: None of the above, but in this section 60J15 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Branching process random graph tree random walk scaling limit alpha-stable tree


Croydon, D. A. Scaling limits for simple random walks on random ordered graph trees. Adv. in Appl. Probab. 42 (2010), no. 2, 528--558. doi:10.1239/aap/1275055241.

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