Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree
type: Item
instance of: scholarly article
title: Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree
zbMATH Open document ID: 1377.60022
DOI: 10.1214/15-AOP1030
published in: The Annals of Probability
publication date: 2017-03-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1407.5162
full work available at URL: https://projecteuclid.org/euclid.aop/1485421327
review text: \textit{R. Pemantle} [Ann. Probab. 19, No. 4, 1559--1574 (1991; Zbl 0758.60010)] showed that the \(n \to \infty\) local limit of the uniform spanning tree on \([-n,n]^2 \cap \mathbb{Z}^2\) (with nearest-neighbour edges) defines a uniform spanning tree (UST) \(\mathcal{U}\) on \(\mathbb{Z}^2\) which is a.s.~a spanning tree, i.e., any two vertices of \(\mathbb{Z}^2\) are connected by a unique path in \(\mathcal{U}\). Denote by \(d_\mathcal{U}\) the intrinsic (shortest-path) metric on the graph \(\mathcal{U}\), and by \(\mu_\mathcal{U}\) the uniform measure that places unit mass at each vertex of \(\mathcal{U}\). Extend the metric space \((\mathcal{U},d_\mathcal{U})\) to a complete and locally compact real tree by adding unit line segments along the edges of \(\mathcal{U}\), and view the measure \(\mu_\mathcal{U}\) as a locally finite Borel measure on the real tree \((\mathcal{U},d_\mathcal{U})\). Moreover, embed this real tree in \(\mathbb{R}^2\) as a spatial tree via the continuous map \(\phi_\mathcal{U} : \mathcal{U} \to \mathbb{R}^2\) which is the identity on the vertices of \(\mathcal{U}\), with linear interpolation along edges. Finally, introduce a root at \(0 \in \mathbb{Z}^2\). The random quintuple \((\mathcal{U},d_\mathcal{U}, \mu_\mathcal{U}, \phi_\mathcal{U},0)\) is the object of the first main result of the paper, which states that its law is tight under rescaling in the appropriate space of measured, rooted spatial trees. Namely, it is shown that the laws \(\mathbf{P}_\delta\), \(\delta \in (0,1)\), associated with \((\mathcal{U}, \delta^\kappa d_\mathcal{U}, \delta^2 \mu_\mathcal{U}, \delta \phi_\mathcal{U}, 0)\) are tight. Here, \(\kappa = 5/4\) is the growth exponent of the loop-erased random walk on \(\mathbb{Z}^2\). The topology preserves information about the intrinsic metric \(d_\mathcal{U}\) and measure \(\mu_\mathcal{U}\), and hence strengthens earlier scaling results of \textit{M. Aizenman} et al. [Random Struct. Algorithms 15, No. 3--4, 319--367…
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60J67
Mathematics Subject Classification ID: 60K37
zbMATH DE Number: 6696264
zbMATH Keywords: uniform spanning tree
zbMATH Keywords: loop-erased random walk
zbMATH Keywords: random walk
zbMATH Keywords: scaling limit
zbMATH Keywords: continuum random tree
author: Martin T. Barlow
author: David A. Croydon
author: Takashi Kumagai
reviewed by: Andrew R. Wade
MaRDI profile type: Publication
arXiv ID: 1407.5162
description: scientific article