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Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree
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Barlow, Martin T., Croydon, David A. and Kumagai, Takashi (2017) Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree. Annals of Probability, 45 (1). pp. 4-55. doi:10.1214/15-AOP1030 ISSN 0091-1798.
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Official URL: http://doi.org/10.1214/15-AOP1030
Abstract
The first main result of this paper is that the law of the (rescaled) two-dimensional uniform spanning tree is tight in a space whose elements are measured, rooted real trees continuously embedded into Euclidean space. Various properties of the intrinsic metrics, measures and embeddings of the subsequential limits in this space are obtained, with it being proved in particular that the Hausdorff dimension of any limit in its intrinsic metric is almost surely equal to 8/5. In addition, the tightness result is applied to deduce that the annealed law of the simple random walk on the two-dimensional uniform spanning tree is tight under a suitable rescaling. For the limiting processes, which are diffusions on random real trees embedded into Euclidean space, detailed transition density estimates are derived.
Item Type: | Journal Article | ||||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||||||
Journal or Publication Title: | Annals of Probability | ||||||||
Publisher: | Institute of Mathematical Statistics | ||||||||
ISSN: | 0091-1798 | ||||||||
Official Date: | 26 January 2017 | ||||||||
Dates: |
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Volume: | 45 | ||||||||
Number: | 1 | ||||||||
Page Range: | pp. 4-55 | ||||||||
DOI: | 10.1214/15-AOP1030 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Open Access Version: |
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