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.

摘要

马尔可夫链遍历的充要条件是该链是不可约且非周期性的。这个结果在有限群上的随机游走的情况下通过关于驱动概率的支持的陈述得到证明：有限群上的随机游走是遍历的当且仅当支持不集中在适当的子群上，也不集中在一个适当的子群上。适当正规子群的陪集。有限群上随机游走的研究自然延伸到有限量子群上的随机游走研究，其中函数代数上的状态起着驱动概率的作用。在驱动态的支持投影上给出了有限量子群上随机游走遍历性的充要条件。