The local limit of the uniform spanning tree on dense graphs

TL;DR

This paper analyzes the local structure of uniform spanning trees in dense graphs, showing convergence to a branching process and establishing sharp bounds on degree distributions in such trees.

Contribution

It characterizes the local limit of uniform spanning trees on dense graphs via graphon convergence and derives sharp bounds on degree distributions.

Findings

Uniform spanning trees locally converge to a multi-type branching process.

Established sharp bounds on the density of leaves and degrees in uniform spanning trees.

Proved that these bounds are optimal.

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 $\{G_n\}$ 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 $1/e-o(1)$, the density of vertices of degree $2$ is at most $1/e+o(1)$ and the density of vertices of degree $k\geq 3$ is at most ${(k-2)^{k-2} \over (k-1)! e^{k-2}} + o(1)$. These bounds are sharp.