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)! e^k-2 + 𝗈(1) . These bounds are sharp.