Local scaling limit of the four-dimensional uniform spanning tree
Abstract
In this paper, we prove that the uniform spanning tree of ℤ4, seen from the origin, converges locally to an infinite spatial Brownian Kesten tree: a Brownian spine onto which independent spatial Brownian trees are grafted at a constant rate. The convergence holds jointly for the intrinsic metric, the volume measure and the Euclidean positions of the vertices, after rescaling these quantities by suitable constant multiples of n−1, n−2(log n)1/3 and n−1/2(log n)−1/6, respectively, under the local rooted spatial Gromov-weak topology. The rescaled space-time occupation measure converges to that of infinite canonical super-Brownian motion. We also identify the limiting law of the rescaled intrinsic-ball volume and obtain sharp first-moment asymptotics.
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BibTeX
@misc{liu2026local,
title = {{Local scaling limit of the four-dimensional uniform spanning tree}},
author = {Runsheng Liu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03582},
eprint = {2611.03582},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03582},
}
RIS
TY - UNPB
TI - Local scaling limit of the four-dimensional uniform spanning tree
AU - Runsheng Liu
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03582
UR - https://arxiv.org/abs/2611.03582
AB - In this paper, we prove that the uniform spanning tree of ℤ^4, seen from the origin, converges locally to an infinite spatial Brownian Kesten tree: a Brownian spine onto which independent spatial Brownian trees are grafted at a constant rate. The convergence holds jointly for the intrinsic metric, the volume measure and the Euclidean positions of the vertices, after rescaling these quantities by suitable constant multiples of n^−1, n^−2(log n)^1/3 and n^−1/2(log n)^−1/6, respectively, under the local rooted spatial Gromov-weak topology. The rescaled space-time occupation measure converges to that of infinite canonical super-Brownian motion. We also identify the limiting law of the rescaled intrinsic-ball volume and obtain sharp first-moment asymptotics.
ER -