Repeat times and a two-weight UST model

Abstract

We study a model of random weighted uniform spanning trees on the complete graph with $n$ vertices, where each edge is assigned a weight of $n^{1+{\hat{I}}^3}$ with probability $1/n$ and $1$ otherwise. Whenever ${\hat{I}}^3$ is large enough, we prove that the diameter of the resulting tree is typically of order $n^{1/3}\log n$, up to a $\log \log n$ correction. Our approach uses estimates on repeat times for selecting components in a critical Erdős-Rényi graph, as well as concentration bounds on the sums of diameters of these components.

32 pages. Comments are welcome!