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January 2017 Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree
M. T. Barlow, D. A. Croydon, T. Kumagai
Ann. Probab. 45(1): 4-55 (January 2017). DOI: 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.

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M. T. Barlow. D. A. Croydon. T. Kumagai. "Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree." Ann. Probab. 45 (1) 4 - 55, January 2017. https://doi.org/10.1214/15-AOP1030

Information

Received: 1 July 2014; Revised: 1 April 2015; Published: January 2017
First available in Project Euclid: 26 January 2017

zbMATH: 1377.60022
MathSciNet: MR3601644
Digital Object Identifier: 10.1214/15-AOP1030

Subjects:
Primary: 60D05, 60G57, 60J60, 60J67, 60K37

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 1 • January 2017
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