This paper considers random walks on a finite group $G$, in which the probability of going from $x$ to $yx, x, y \in G$, depends only on $y$. The main results concern the distribution of the number of steps it takes to reach a particular element of $G$ if one starts with the uniform distribution on $G$. These results answer some random sorting questions. They are attained by applications of group representation theory.
"Random Shuffles and Group Representations." Ann. Probab. 13 (1) 154 - 178, February, 1985. https://doi.org/10.1214/aop/1176993073