November 2021 Scaling limits of the three-dimensional uniform spanning tree and associated random walk
O. Angel, D. A. Croydon, S. Hernandez-Torres, D. Shiraishi
Author Affiliations +
Ann. Probab. 49(6): 3032-3105 (November 2021). DOI: 10.1214/21-AOP1523


We show that the law of the three-dimensional uniform spanning tree (UST) is tight under rescaling in a space whose elements are measured, rooted real trees, continuously embedded into Euclidean space. We also establish that the relevant laws actually converge along a particular scaling sequence. The techniques that we use to establish these results are further applied to obtain various properties of the intrinsic metric and measure of any limiting space, including showing that the Hausdorff dimension of such is given by 3/β, where β1.624 is the growth exponent of three-dimensional loop-erased random walk. Additionally, we study the random walk on the three-dimensional uniform spanning tree, deriving its walk dimension (with respect to both the intrinsic and Euclidean metric) and its spectral dimension, demonstrating the tightness of its annealed law under rescaling, and deducing heat kernel estimates for any diffusion that arises as a scaling limit.

Funding Statement

D. Croydon would like to acknowledge the support of a JSPS Grant-in-Aid for Research Activity Start-up, 18H05832, a JSPS Grant-in-Aid for Scientific Research (C), 19K03540, and the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University. S. Hernandez-Torres would like to acknowledge the support of a fellowship from the Mexican National Council for Science and Technology (CONACYT). D. Shiraishi is supported by a JSPS Grant-in-Aid for Early-Career Scientists, 18K13425 and JSPS KAKENHI Grant Numbers 17H02849 and 18H01123.


The authors would like to thank Russell Lyons for suggesting numerous corrections, and an anonymous referee for their insightful referee report. In particular, for the referee’s detailed description of how Kozma’s result from [31] could be extended (see Remark 1.2 for further comments).


Download Citation

O. Angel. D. A. Croydon. S. Hernandez-Torres. D. Shiraishi. "Scaling limits of the three-dimensional uniform spanning tree and associated random walk." Ann. Probab. 49 (6) 3032 - 3105, November 2021.


Received: 1 March 2020; Revised: 1 December 2020; Published: November 2021
First available in Project Euclid: 7 December 2021

MathSciNet: MR4348685
zbMATH: 1486.60019
Digital Object Identifier: 10.1214/21-AOP1523

Primary: 60D05 , 60G57 , 60K37

Keywords: Random walk , Scaling limit , Transition density , Uniform spanning tree

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 6 • November 2021
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