Local Limit of Weighted Spanning Trees on Networks
Analysis
This paper investigates the local behavior of weighted spanning trees (WSTs) on high-degree, almost regular or balanced networks. It generalizes previous work and addresses a gap in a prior proof. The research is motivated by studying an interpolation between uniform spanning trees (USTs) and minimum spanning trees (MSTs) using WSTs in random environments. The findings contribute to understanding phase transitions in WST properties, particularly on complete graphs, and offer a framework for analyzing these structures without strong graph assumptions.
Key Takeaways
“The paper proves 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.”