Local scaling limit of the four-dimensional uniform spanning tree

Abstract

In this paper, we prove that the uniform spanning tree of $\mathbb Z^4$, 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.