Local scaling limit of the four-dimensional uniform spanning tree

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.