The diameter of random spanning trees interpolating between the UST and the MST of the complete graph

We introduce ${\mathsf{W}\mathsf{S}\mathsf{T}}^{{\beta }_n}(K_n)$ as the weighted spanning tree of the complete graph $K_n$ w.r.t. the random electric network of conductances $\{\exp (-{\beta }_n{U}_e){\}}_{e\in E(K_n)}$ with $\mathrm{U}\mathrm{n}\mathrm{i}\mathrm{f}[0,1]$ i.i.d. $U_e$’s. Moving from ${\beta }_n\equiv 0$ to faster and faster growing ${\beta }_n$’s, the model interpolates between the \emph{uniform} and the \emph{minimum} spanning trees: ${\mathsf{W}\mathsf{S}\mathsf{T}}^0(K_n)=\mathsf{U}\mathsf{S}\mathsf{T}(K_n)$, and there are phase transitions for ${\mathsf{W}\mathsf{S}\mathsf{T}}^{{\beta }_n}(K_n)$ behaving more and more like $\mathsf{M}\mathsf{S}\mathsf{T}(K_n)$: - around ${\beta }_n=n^{3+o(1)}$ regarding the agreement of the two standard algorithms generating these models : Aldous-Broder and Prim’s invasion algorithms, - around ${\beta }_n=n^{2+o(1)}$ regarding the models consisting of exactly the same edges, and - around ${\beta }_n=n^{1+o(1)}$ regarding the expected total length $\mathbb{E}\left[{\sum }_{e\in {\mathsf{W}\mathsf{S}\mathsf{T}}^{{\beta }_n}(K_n)}{U}_e\right]$. But most importantly, we study the global geometry of the model: we prove that the typical diameter of ${\mathsf{W}\mathsf{S}\mathsf{T}}^{{\beta }_n}(K_n)$ grows like $\mathrm{\Theta }(n^{1\mathrm{/}3})$ for ${\beta }_n\ge n^{4\mathrm{/}3+o(1)}$ likewise the $\mathsf{M}\mathsf{S}\mathsf{T}(K_n)$ case, and it grows like $\mathrm{\Theta }(n^{1\mathrm{/}2})$ for ${\beta }_n\le n^{1+o(1)}$ similarly to the $\mathsf{U}\mathsf{S}\mathsf{T}(K_n)$ case. For ${\beta }_n=n^{\alpha }$ with $1<\alpha <4\mathrm{/}3$, the behavior of the typical diameter is a more delicate open question, but we conjecture that its exponent strictly between 1/2 and 1/3.