Green functions for killed random walks in the Weyl chamber of Sp(4)

TL;DR

This paper analyzes a family of killed random walks in the Weyl chamber of Sp(4), deriving exact asymptotics for Green functions and absorption probabilities, with connections to reflection groups and quantum walks.

Contribution

It introduces a new family of random walks associated with reflection groups of various orders and provides explicit asymptotic results for their Green functions and absorption probabilities.

Findings

Exact asymptotics of Green functions along all infinite paths.

Asymptotic behavior of absorption probabilities at the boundary.

Identification of a process related to quantum random walks on Sp(4).

Abstract

We consider a family of random walks killed at the boundary of the Weyl chamber of the dual of $\rm{Sp}(4)$, which in addition satisfies the following property: for any $n\geq 3$, there is in this family a walk associated with a reflection group of order $2n$. Moreover, the case $n=4$ corresponds to a process which appears naturally by studying quantum random walks on the dual of $\rm{Sp}(4)$. For all the processes belonging to this family, we find the exact asymptotic of the Green functions along all infinite paths of states as well as that of the absorption probabilities along the boundaries.