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Scaling limits for simple random walks on random ordered graph trees
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Croydon, David A. (2010) Scaling limits for simple random walks on random ordered graph trees. Advances in Applied Probability, Vol.42 (No.2). pp. 528-558. doi:10.1239/aap/1275055241 ISSN 0001-8678.
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Official URL: http://dx.doi.org/10.1239/aap/1275055241
Abstract
Consider a family of random ordered graph trees (T-n)(n >= 1), where T-n 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 -> infinity, 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 resealed to converge to the Brownian motion on a related alpha-stable tree.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||
Journal or Publication Title: | Advances in Applied Probability | ||||
Publisher: | Applied Probability Trust | ||||
ISSN: | 0001-8678 | ||||
Official Date: | June 2010 | ||||
Dates: |
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Volume: | Vol.42 | ||||
Number: | No.2 | ||||
Number of Pages: | 31 | ||||
Page Range: | pp. 528-558 | ||||
DOI: | 10.1239/aap/1275055241 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
Data sourced from Thomson Reuters' Web of Knowledge
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