Semifinite harmonic functions on branching graphs

TL;DR

This paper provides a combinatorial approach to classifying semifinite harmonic functions on branching graphs, inspired by operator algebra methods and the theory of infinite symmetric group representations.

Contribution

It rephrases and simplifies Wassermann's algebraic method for classifying harmonic functions using combinatorial objects.

Findings

Classified semifinite indecomposable harmonic functions on certain branching graphs.

Reformulated operator algebra approach in purely combinatorial terms.

Enhanced understanding of harmonic functions related to the infinite symmetric group.

Abstract

We study semifinite harmonic functions on arbitrary branching graphs. We give a detailed exposition of an algebraic method which allows one to classify semifinite indecomposable harmonic functions on some multiplicative branching graphs. This method was proposed by A. Wassermann in terms of operator algebras, while we rephrase, clarify, and simplify the main arguments, working only with combinatorial objects. This work was inspired by the theory of traceable factor representations of the infinite symmetric group $S(\infty)$.