Monopoles, dipoles, and harmonic functions on Bratteli diagrams

TL;DR

This paper develops explicit formulas for special functions on electrical networks and explores harmonic functions on Bratteli diagrams, providing algorithmic descriptions and conditions for finite energy in specific classes.

Contribution

It introduces a method to explicitly compute harmonic functions, monopoles, and dipoles on Bratteli diagrams, linking network structure to function properties.

Findings

Algorithmic description of harmonic functions on Bratteli diagrams

Conditions for finite energy harmonic functions on specific diagram classes

Explicit formulas for monopoles and dipoles in electrical networks

Abstract

In our study of electrical networks we develop two themes: finding explicit formulas for special classes of functions defined on the vertices of a transient network, namely monopoles, dipoles, and harmonic functions. Secondly, our interest is focused on the properties of electrical networks supported on Bratteli diagrams. We show that the structure of Bratteli diagrams allows one to describe algorithmically harmonic functions as well as monopoles and dipoles. We also discuss some special classes of Bratteli diagrams (stationary, Pascal, trees), and we give conditions under which the harmonic functions defined on these diagrams have finite energy.