Reference requested: Random walk on groups I am looking for a good reference to learn about random walks on groups (either finite groups or Lie groups). Ideally, I would like a reference for general theory of random walks on groups that is self-contained in terms of the probability theory it relies on, while remaining condensed and precise. As an analogy, something like Atiyah-Macdonald's book in commutative algebra (i.e. short, but very precise and somehow complete) would be great.  
Thank you very much.     
 A: You could also consider the book
Wolfgang Woess, Random Walks on Infinite Graphs and Groups, Cambridge University Press, 2000.
This is a good book containing a lot of information. However, it mainly focusses on infinite discrete groups and it might be not so easy for self-study. However, I would certainly recommend to have a look at it.
A: How about these two books for a start?
Tullio Ceccherini-Silberstein, Fabio Scarabotti, and Filippo Tolli. Harmonic analysis on finite groups. Representation theory, Gelfand pairs and Markov chains. Cambridge Studies in Advanced Mathematics 108. Cambridge: Cambridge University Press, 2008.
Persi Diaconis. Group representations in probability and statistics. IMS Lecture Notes-Monograph Series, 11. Hayward, CA: Institute of Mathematical Statistics, 1988.
Referenced added: If you are interested in Lie groups, you could also start with this survey by Emmanuel Breuillard: http://www.math.u-psud.fr/~breuilla/part0gb.pdf
A: My book http://link.springer.com/book/10.1007%2F978-1-4614-0776-8 on group representation theory has a chapter on this. 
A: To focusing on special groups. For instance random walks on solvable Lie groups. The best reference is the doctoral thesis of Tianyi Zheng in Cornell University. You can find the pdf fine here. Also on lie groups you can find this nice paper here. About random walks on nilpotent lie groups you can find the best reference in a doctoral thesis here
About random walks on finite groups, the best can be the note of Laurent Saloff-Coste entitled Random Walks on Finite Groups. You can find the pdf file  here 
A: I did my M.Sc thesis on random walks on finite groups and you might be interested in the proof of Theorem 1.3.2 in particular.
When I was looking for the proof of this theorem, almost all references refer
to the proof in older hard-to-source references — if at all. A proof outline is given
by Fountoulakis in his lecture notes but here a full proof is given.
