The first main result of this paper is that the law of the (rescaled) two-dimensional uniform
spanning tree is tight in a space whose elements are measured, rooted real trees
continuously embedded into Euclidean space. Various properties of the intrinsic metrics,
measures and embeddings of the subsequential limits in this space are obtained, with it
being proved in particular that the Hausdorff dimension of any limit in its intrinsic metric is
almost surely equal to 8/5. In addition, the tightness result is applied to deduce that the …

… A subgraph of the lattice is a spanning tree of ⇥n if it connects all vertices, no cycles. Let
U(n) be a spanning tree of ⇥n selected uniformly at random from all possibilities. … So we
would obtain a loop around 0 ― which is impossible since U is a tree. …