Convergence to the boundary for random walks on discrete quantum groups and monoidal categories

TL;DR

This paper investigates how random walks on discrete quantum groups and related monoidal categories converge to their boundaries, establishing results for specific cases and defining the Martin boundary in categorical contexts.

Contribution

It introduces a categorical framework for the Martin boundary, demonstrating the stability of boundary convergence under monoidal equivalence for quantum groups.

Findings

Convergence to the boundary is proven for random walks on itSU_q(2).

A categorical definition of the Martin boundary is provided and shown to align with quantum group cases.

Boundary convergence remains stable under monoidal equivalence of quantum groups.

Abstract

We study the problem of convergence to the boundary in the setting of random walks on discrete quantum groups. Convergence to the boundary is established for random walks on $\hat{\textrm{SU}_q(2)}$. Furthermore, we will define the Martin boundary for random walks on C$^*$-tensor categories and give a formulation for convergence to the boundary for such random walks. These categorical definitions are shown to be compatible with the definitions in the quantum group case. This implies that convergence to the boundary for random walks on quantum groups is stable under monoidal equivalence.