Random Walks on Wreath Products of Groups

Abstract

We bound the rate of convergence to uniformity for a certain random walk on the complete monomial groups G≀S _n for any group G. Specifically, we determine that ${\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}$ n log n+$\frac{1}{4}$ n log (|G|−1|) steps are both necessary and sufficient for ℓ² distance to become small. We also determine that ${\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}$ n log n steps are both necessary and sufficient for total variation distance to become small. These results provide rates of convergence for random walks on a number of groups of interest: the hyperoctahedral group ℤ₂≀S _n , the generalized symmetric group ℤ _m ≀S _n , and S _m ≀S _n . In the special case of the hyperoctahedral group, our random walk exhibits the “cutoff phenomenon.”