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 $n^{1+γ}$ with probability $1/n$ and $1$ otherwise. Whenever $γ$ 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.