The Local Limit of the Uniform Spanning Tree on Dense Graphs

Jan Hladký, Asaf Nachmias*, Tuan Tran

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

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 { Gn} is a sequence of such graphs converging to a graphon W, then the uniform spanning tree of Gn 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.

Original languageEnglish
Pages (from-to)502-545
Number of pages44
JournalJournal of Statistical Physics
Volume173
Issue number3-4
DOIs
StatePublished - 1 Nov 2018

Keywords

  • Benjamini-Schramm convergence
  • Branching process
  • Graph limits
  • Graphon
  • Uniform spanning tree

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