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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 aremeasured, rooted real trees continuously embedded into Euclidean space. Variousproperties of the intrinsic metrics, measures and embeddings of thesubsequential limits in this space are obtained, with it being proved inparticular that the Hausdorff dimension of any limit in its intrinsic metric isalmost surely equal to $8/5$. In addition, the tightness result is applied todeduce that the annealed law of the simple random walk on the two-dimensionaluniform spanning tree is tight under a suitable rescaling. For the limitingprocesses, which are diffusions on random real trees embedded into Euclideanspace, detailed transition density estimates are derived.

Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree