I am about to finish working through Williams's Probability With Martingales. I have studied analysis up to the first five chapters of Folland's text but have not studied any combinatorics yet.

It seems like 'combinatorial' probability topics like percolation, probability on graphs and networks, finite Markov chains and random walks are currently very active and I would like to be able to read the current research in at least some of these areas.

While I can find many interesting texts on Amazon etc. I am not sure how well they reflect current work.

I would greatly appreciate a reading list or learning roadmap for this area.

[I am cross-posting this from Math.SE where it did not get an answer despite a bounty.]


3 Answers 3


Geoffrey Grimmett's Probability on Graphs is an excellent introduction to a variety of current active research areas in discrete probability theory, and is probably at about the level you want (you may find that you need to consult other books if there is background you need to fill in, but that isn't a bad thing). It also has quite a number of very stimulating exercises. It's available free from his web site.


Some suggestions:

Levin, Peres and Wilmer have a very nice (allegedly undergraduate) textbook on Mixing times of Markov Chains. This contains many advanced topics.

Or you could try the Lyons (with Peres) book Probability on trees and networks.


This is a large topic and if you want an actual roadmap to current research then I think you'll need to narrow your focus a little.

Having said that, I don't think it can hurt to familiarize yourself with the basics of random graphs. I would recommend Béla Bollobás's textbook. If you then want to move on to percolation theory, Bollobás also has a book on percolation (with Riordan). I'm not very familiar with it (or with percolation theory for that matter), but I trust the authors' taste and expository skills so I expect that it is a good introduction to the subject, and it would be the book I would try first if I wanted to venture into that area.

For the Markov chain angle, I find Olle Haggstrom's Finite Markov Chains and Algorithmic Applications to be a quick and painless introduction.


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