Repeat times and a two-weight UST model

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 $n^{1+\gamma }$ with probability $1\mathrm{/}n$ and $1$ otherwise. Whenever $\gamma$ is large enough, we prove that the diameter of the resulting tree is typically of order $n^{1\mathrm{/}3}\log n$, 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.