A note on the Martin boundary of the simple random walk on an example of directed graph

TL;DR

This paper investigates the Martin boundary of a simple random walk on a specific directed graph, demonstrating that it is trivial with no non-constant positive harmonic functions, supported by detailed Green function estimates.

Contribution

It provides a detailed analysis of the Martin boundary for a particular directed graph, showing its triviality, which is a novel example in the study of harmonic functions on graphs.

Findings

Martin boundary is trivial for the studied directed graph

No non-constant positive harmonic functions exist in this case

Green function estimates are crucial for the proof

Abstract

In this note, we study the Martin boundary of the simple random walk on an example of directed graph. More precisely, the Martin boundary is shown to be trivial, i.e. there is no non-constant positive harmonic functions. The proof involves fine estimates of the Green function which are summarized in this note.