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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.

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Furstenberg's classical paper about the product of random matrices is very-well written... If I'm not mistaken Varju is currently writing a book on this subject. –  Asaf Feb 21 '14 at 15:10
Since the question was asked there have been some developments, especially Yves Benoist and Jean-Francois Quint have released their book "Random Walks on Reductive Groups", available freely from Jean-Francois' website. –  Asaf Jun 24 at 19:05

5 Answers 5

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

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Both excellent books. –  Jp McCarthy Feb 21 '14 at 10:13

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.

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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

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There is a problem with your links in the first paragraph. The first two links go to Breuillard's survey and the third goes to a paper by Astashkevich and Pak, not to a thesis. –  UwF Feb 21 '14 at 10:17
I edited my answer, thanks –  Hassan Jolany Feb 21 '14 at 11:34

My book http://link.springer.com/book/10.1007%2F978-1-4614-0776-8 on group representation theory has a chapter on this.

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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.

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