Random spanning trees in random environment

We introduce a new spanning tree model called the random spanning tree in random environment (RSTRE), which interpolates between the uniform spanning tree and the minimum spanning tree as the inverse temperature (disorder strength) $\beta$ varies. On the complete graph with $n$ vertices and i.i.d.\ uniform disorder variables on the edges, we identify: (1) a low disorder regime with $\beta \le Cn\mathrm{/}\log n$, where the diameter of the random spanning tree is typically of order $n^{1\mathrm{/}2}$, the same as for the uniform spanning tree; (2) a high disorder regime with $\beta \ge n^{4\mathrm{/}3}\log n$, where the diameter is typically of order $n^{1\mathrm{/}3}$, the same as for the minimum spanning tree. We conjecture that for $\beta =n^{\alpha }$ with $\alpha \in (1,4\mathrm{/}3)$, the diameter is of order $n^{\gamma +o(1)}$ for some $\gamma =\gamma (\alpha )$ strictly between $1\mathrm{/}2$ and $1\mathrm{/}3$.