Let G be a finite group. The commuting chain on G moves from an element x to y by selecting y uniformly amongst those which commute with x. The t-step transition probabilities of this chain converge to a distribution uniform on the conjugacy classes of G. We provide upper and lower bounds for the mixing time of this chain on a CA group (groups with a “nice” commuting structure) and show that cutoff does not occur for many of these chains. We also provide a formula for the characteristic polynomial of the transition matrix of this chain. We apply our general results to explicitly study the chain on several sequences of groups, such as general linear groups, Heisenberg groups, and dihedral groups. The commuting chain is a specific case of a more general family of chains known as Burnside processes. Few instances of the Burnside processes have permitted careful analysis of mixing. We present some of the first results on mixing for the Burnside process where the state space is not fully specified (i.e., not for a particular group). Our upper bound shows our chain is rapidly mixing, a topic of interest for Burnside processes.

令G为有限群。G上的通勤链通过在与x通勤的元素中均匀地选择y来从元素x移动到y。该链的t步转移概率收敛到G的共轭类上的均匀分布. 我们提供了该链在 CA 组（具有“良好”通勤结构的组）上的混合时间的上限和下限，并表明许多这些链不会发生截止。我们还提供了该链的转移矩阵的特征多项式的公式。我们应用我们的一般结果来明确研究几个群序列上的链，例如一般线性群、海森堡群和二面体群。通勤链是称为 Burnside 流程的更一般的链族的一个特定案例。很少有 Burnside 过程允许对混合进行仔细分析。我们展示了一些关于 Burnside 过程混合的初步结果，其中状态空间没有完全指定（即，不是针对特定组）。我们的上限表明我们的链正在快速混合，