Green functions and Martin compactification for killed random walks related to SU(3)

TL;DR

This paper analyzes killed random walks in the quarter plane with specific properties, determining their Green function asymptotics and showing the Martin compactification reduces to a point.

Contribution

It provides the first detailed asymptotic analysis of Green functions for these specific random walks and establishes the Martin compactification as a point.

Findings

Green functions asymptotics along all paths are characterized.

Martin compactification is shown to be the one-point compactification.

Harmonic polynomial of degree three plays a key role.

Abstract

We consider the random walks killed at the boundary of the quarter plane, with homogeneous non-zero jump probabilities to the eight nearest neighbors and drift zero in the interior, and which admit a positive harmonic polynomial of degree three. For these processes, we find the asymptotic of the Green functions along all infinite paths of states, and from this we deduce that the Martin compactification is the one-point compactification.