The local limit of weighted spanning trees on balanced networks

TL;DR

The paper analyzes the local limits of weighted spanning trees on high-degree, almost regular networks, and shows that the local limit is the Poisson(1) Galton–Watson tree conditioned to survive forever. It extends previous work to the weighted setting and includes a correction to a gap in prior proofs, while also introducing and studying the one-parameter environment $\mathsf{WST}^{\beta}(G)$ that interpolates between UST and MST. A phase transition is established for the local limit and edge overlaps in high-degree almost balanced graphs, with the critical scale around $\beta_n$ comparable to the average degree times a slowly varying factor, and results extend to complete graphs where explicit asymptotics are obtained. The paper also develops a robust deterministic framework and extends it to random i.i.d. environments, showing that the local limit, edge overlaps, and total length exhibit regime-dependent behavior, including precise results for complete graphs and general high-degree sequences. Overall, the work provides a unified, transferable framework for understanding local limits of spanning-tree models in both deterministic and random settings, with concrete phase-transition phenomena and links to MST/UST behavior.

Abstract

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 $\{\mathsf{WST}^β(G)\}_{β\in [0, \infty)}$ 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, Kús24], and the phases of several properties have been determined on the complete graphs. We show a phase transition of $\mathsf{WST}^{β_n}(G_n)$ 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].

Theorems & Definitions (69)

• Definition 1.1
• Definition 1.2
• Theorem 1.1
• Definition 1.3
• Definition 1.4
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4: Phase transition of the local limit of $\mathsf{WST}^{\beta_n}(G_n)$ on high degree almost balanced graphs
• Definition
• Theorem 1.5