Metric Graphs with Totally Disconnected Boundary

TL;DR

This paper develops boundary analysis for infinite weighted graphs with compact completions and totally disconnected boundaries, constructing Haar bases, Hilbert spaces, and analyzing harmonic exit measures.

Contribution

It introduces a novel boundary analysis framework for infinite weighted graphs with totally disconnected boundaries, including Haar bases and harmonic measure construction.

Findings

Constructed generalized Haar bases on graph boundaries

Developed Hilbert spaces of boundary functions

Analyzed exit measures using harmonic functions

Abstract

Boundary analysis is developed for a rich class of generally infinite weighted graphs with compact metric completions. These graph completions have totally disconnected boundaries. The classical notion of $\epsilon$-components and the existence of suitable measures are used to construct generalized Haar bases and Hilbert spaces of functions on the boundaries. Suitable exit measures are constructed and analyzed using harmonic functions.