The ergodic theorem for random walks on finite quantum groups

Abstract

Necessary and sufficient conditions for a Markov chain to be ergodic are that the chain is irreducible and aperiodic. This result is manifest in the case of random walks on finite groups by a statement about the support of the driving probability: a random walk on a finite group is ergodic if and only if the support is not concentrated on a proper subgroup, nor on a coset of a proper normal subgroup. The study of random walks on finite groups extends naturally to the study of random walks on finite quantum groups, where a state on the algebra of functions plays the role of the driving probability. Necessary and sufficient conditions for ergodicity of a random walk on a finite quantum group are given on the support projection of the driving state.

Keywords:

• Ergodicity
• finite quantum groups
• random walks

2020 Mathematics Subject Classification:

• 46L53 (60J05
• 20G42)

Acknowledgments

I would like to thank Uwe Franz; much progress on this problem was achieved during a (very enjoyable) May 2019 visit to Uwe at the Laboratoire de mathématiques de Besançon (${\text{Lm}}^{\mathrm{B}}$), France.

Additional information

Funding

This trip was financially supported by Lm^B, and also Cork Institute of Technology.