Mixing Times for Markov Chains on Wreath Products and Related Homogeneous Spaces

Abstract

We develop a method for analyzing the mixing times for a quite general class of Markov chains on the complete monomial group $G \wr S_n$ and a quite general class of Markov chains on the homogeneous space $(G\wr S_n) / (S_r\times S_{n-r})$. We derive an exact formula for the $L^2$ distance in terms of the $L^2$ distances to uniformity for closely related random walks on the symmetric groups $S_j$ for $1 \leq j \leq n$ or for closely related Markov chains on the homogeneous spaces $S_{i+j}/ (S_i~\times~S_j)$ for various values of $i$ and $j$, respectively. Our results are consistent with those previously known, but our method is considerably simpler and more general.