## The Annals of Probability

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

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

M. T. Barlow, D. A. Croydon, and T. Kumagai

#### 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**

Received: July 2014

Revised: April 2015

First available in Project Euclid: 26 January 2017

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/15-AOP1030

**Mathematical Reviews number (MathSciNet)**

MR3601644

**Zentralblatt MATH identifier**

1377.60022

**Subjects**

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G57: Random measures 60J60: Diffusion processes [See also 58J65] 60J67: Stochastic (Schramm-)Loewner evolution (SLE) 60K37: Processes in random environments

**Keywords**

Uniform spanning tree loop-erased random walk random walk scaling limit continuum random tree

#### 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