Research Paper#Graph Theory, Measure Theory, Functional Analysis🔬 ResearchAnalyzed: Jan 3, 2026 17:08
Radon-Nikodym Topography of Acyclic Measured Graphs
Published:Dec 31, 2025 09:20
•1 min read
•ArXiv
Analysis
This paper investigates the geometric and measure-theoretic properties of acyclic measured graphs, focusing on the relationship between their 'topography' (geometry and Radon-Nikodym cocycle) and properties like amenability and smoothness. The key contribution is a characterization of these properties based on the number and type of 'ends' in the graph, extending existing results from probability-measure-preserving (pmp) settings to measure-class-preserving (mcp) settings. The paper introduces new concepts like 'nonvanishing ends' and the 'Radon-Nikodym core' to facilitate this analysis, offering a deeper understanding of the structure of these graphs.
Key Takeaways
- •Introduces a topographic analysis of acyclic measured graphs.
- •Characterizes amenability and smoothness based on the number of nonvanishing ends.
- •Extends Adams dichotomy from pmp to mcp settings.
- •Introduces the concept of Radon-Nikodym core.
- •Provides a surprising topographic characterization of when certain functions are essentially one-ended.
Reference
“An acyclic mcp graph is amenable if and only if a.e. component has at most two nonvanishing ends, while it is nowhere amenable exactly when a.e. component has a nonempty perfect (closed) set of nonvanishing ends.”