Random walk on group is defined in the following way as a Markov chain.
A theorem says the uniform distribution is stationary for all random walk on groups.
If the random walk is irreducible, for example, on a cyclic group $\Bbb Z_n$, then the stationary distribution is unique and hence uniform. My question is, is there any good example of reducible random walk on groups so that the uniform distribution is not the only stationary distribution?
It is really hard for me to imagine such a reducible random walk. All examples for random walk on groups given in the textbook so far are irreducible. Hope someone can help. Thank you!
It is not quite true that any random walk on $\mathbb{Z}_n$ is irreducible. This will depend on the increment distribution $\mu$ as well.
For example, on $\mathbb{Z}_4$ if you take $\mu(g)=\delta_{g+2}$ then there are two communicating classes $\{0,2\}$ and $\{1,3\}.$ There are lots of different stationary distributions for this random walk.