Intermediate-regime diameter interpolation for the exponential-weight RSTRE

Characterize the intermediate regime of the parameter β for the exponential-weight random spanning tree in random environment on the complete graph, and determine whether its typical diameter smoothly interpolates between the uniform spanning tree scale n^{1/2} and the minimum spanning tree scale n^{1/3}.

Background

The paper reviews prior results for a random spanning tree in a random environment on the complete graph with edge weights of the form exp(-βω_e). For β at most n{1-o(1)}, the typical diameter is of order n{1/2}, whereas for β at least n{4/3+o(1)}, it is of order n{1/3}. The cited conjecture concerns the unresolved behavior between these two regimes and asks whether the diameter varies smoothly rather than undergoing an abrupt transition.

References

It is conjectured, see Conjecture 1.3, that there is some intermediate regime for the choice of parameter $\beta$ such that the diameter smoothly interpolates between the two power laws $\sqrt{n}$ and $n{1/3}$ corresponding to the UST and MST, respectively.

— Repeat times and a two-weight UST model  (2512.21977 - Ambroggio et al., 26 Dec 2025) in Section 1, Introduction