Local limits of random spanning trees in random environment

We study the edge overlap and local limit of the random spanning tree in random environment (RSTRE) on the complete graph with $n$ vertices and weights given by $\exp (-\beta {\omega }_e)$ for ${\omega }_e$ uniformly distributed on $[0,1]$. We show that for $\beta$ growing with $\beta =o(n\mathrm{/}\log n)$, the edge overlap is $(1+o(1))\beta$, while for $\beta$ much larger than $n{\log }^{2}n$, the edge overlap is $(1-o(1))n$. Furthermore, there is a transition of the local limit around $\beta =n$. When $\beta =o(n\mathrm{/}\log n)$ the RSTRE locally converges to the same limit as the uniform spanning tree, whereas for $\beta$ larger than $n{\log }^{\lambda }n$, where $\lambda =\lambda (n)\to \mathrm{\infty }$ arbitrarily slowly, the local limit of the RSTRE is the same as that of the minimum spanning tree.