The Local Limit of the Uniform Spanning Tree on Dense Graphs

摘要

Let G be a connected graph in which almost all vertices have linear degrees and let $$mathcal {T}$$ 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 $$mathcal {T}$$ T is isomorphic to F . We deduce from this that if $${G_n}$$ { G n } is a sequence of such graphs converging to a graphon W , then the uniform spanning tree of $$G_n$$ 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 $$e^{-1}-mathsf {o}(1)$$ e - 1 - o ( 1 ) , the density of vertices of degree 2 is at most $$e^{-1}+mathsf {o}(1)$$ e - 1 + o ( 1 ) and the density of vertices of degree $$kgeqslant 3$$ k ? 3 is at most $${(k-2)^{k-2} over (k-1)! e^{k-2}} + mathsf {o}(1)$$ ( k - 2 ) k - 2 ( k - 1 ) ! e k - 2 + o ( 1 ) . These bounds are sharp.