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.
Cite
@article{arxiv.2004.01234,
title = {The Ergodic Theorem for Random Walks on Finite Quantum Groups},
author = {J. P. McCarthy},
journal= {arXiv preprint arXiv:2004.01234},
year = {2021}
}
Comments
27 pages, v2: revisions to some sections, a significant reduction in length in others