## Abstract

The first main result of this paper is that the law of the (rescaled) twodimensional 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.

Original language | English |
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Pages (from-to) | 4-55 |

Number of pages | 52 |

Journal | Annals of Probability |

Volume | 45 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2017 |

Externally published | Yes |

## Keywords

- Continuum random tree
- Loop-erased random walk
- Random walk
- Scaling limit
- Uniform spanning tree

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty