The uniform spanning tree in 4 dimensions

A uniform spanning tree of $\mathbb{Z}^4$ can be thought of as the "uniform measure" on trees of $\mathbb{Z}^4$. The past of 0 in the uniform spanning tree is the finite component that is disconnected from infinity when 0 is deleted from the tree. We establish the logarithmic corrections to the probabilities that the past contains a path of length $n$, that it has volume at least $n$ and that it reaches the boundary of the box of side length $n$ around 0. Dimension 4 is the upper critical dimension for this model in the sense that in higher dimensions it exhibits "mean-field" critical behaviour. An important part of our proof is the study of the Newtonian capacity of a loop erased random walk in 4 dimensions. This is joint work with Tom Hutchcroft.