## The Annals of Probability

### Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree

#### 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.

#### Article information

Source
Ann. Probab., Volume 45, Number 1 (2017), 4-55.

Dates
Revised: April 2015
First available in Project Euclid: 26 January 2017

https://projecteuclid.org/euclid.aop/1485421327

Digital Object Identifier
doi:10.1214/15-AOP1030

Mathematical Reviews number (MathSciNet)
MR3601644

Zentralblatt MATH identifier
1377.60022

#### Citation

Barlow, M. T.; Croydon, D. A.; Kumagai, T. Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree. Ann. Probab. 45 (2017), no. 1, 4--55. doi:10.1214/15-AOP1030. https://projecteuclid.org/euclid.aop/1485421327

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