The GHP Scaling Limit of Uniform Spanning Trees in High Dimensions.

We show that the Brownian continuum random tree is the Gromov-Hausdorff-Prohorov scaling limit of the uniform spanning tree on high-dimensional graphs including the d -dimensional torus Z n d with d > 4 , the hypercube { 0 , 1 } n , and transitive expander graphs. Several corollaries for associated quantities are then deduced: convergence in distribution of the rescaled diameter, height and simple random walk on these uniform spanning trees to their continuum analogues on the continuum random tree.