The local limit of weighted spanning trees on balanced networks

We prove that the local limit of the weighted spanning trees on any simple connected high degree almost regular sequence of electric networks is the Poisson(1) branching process conditioned to survive forever, by generalizing [NP22] and closing a gap in their proof. We also study the local statistics of the WST's on high degree almost balanced sequences, which is interesting even for the uniform spanning trees. Our motivation comes from studying an interpolation $\{\mathsf{WST}^{\beta}(G)\}_{\beta\in [0, \infty)}$ between UST(G) and MST(G) by WST's on a one-parameter family of random environments. This model has recently been introduced in [MSS24, K\'us24], and the phases of several properties have been determined on the complete graphs. We show a phase transition of $\mathsf{WST}^{\beta_n}(G_n)$ regarding the local limit and expected edge overlaps for high degree almost balanced graph sequences $G_n$, without any structural assumptions on the graphs; while the expected total length is sensitive to the global structure of the graphs. Our general framework results in a better understanding even in the case of complete graphs, where it narrows the window of the phase transition of [Mak24].