The Local Limit of the Uniform Spanning Tree on Dense Graphs

Let G be a connected graph in which almost all vertices have linear degrees and let T be a uniform spanning tree of G. For any fixed rooted tree F of height r we compute the asymptotic density of vertices v for which the r-ball around v in T is isomorphic to F. We deduce from this that if { G_n} is a sequence of such graphs converging to a graphon W, then the uniform spanning tree of G_n locally converges to a multi-type branching process defined in terms of W. As an application, we prove that in a graph with linear minimum degree, with high probability, the density of leaves in a uniform spanning tree is at least e^{- 1}- o(1) , the density of vertices of degree 2 is at most e^{- 1}+ o(1) and the density of vertices of degree k⩾ 3 is at most (k-2)k-2(k-1)!ek-2+o(1). These bounds are sharp.