Markov chain on groups

This is a classical question which is now well understood, I believe. Start with this famous paper: http://tinyurl.com/yfyrg63 Although this question is really not about standard r.w. on groups, a survey by Saloff-Coste gives an excellent introduction and literature review: www.math.cornell.edu/~lsc/rwfg.pdf

I'm not sure, but I'll bet you can find the answer in the recent book Markov Chains and Mixing Times by Peres, Levin and Wilmer.

I second Tom's suggestion. Another place to look is the not-yet-published book Reversible Markov Chains and Random Walks on Graphs by Aldous and Fill.