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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.

Rushabh Mehta
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Jyotirmoy Bhattacharya
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    A start perhaps: http://math.stackexchange.com/questions/542135/are-there-open-questions-in-markov-chains –  Sep 28 '14 at 22:52
  • As there wasn't an answer here despite a bounty, I am not posting this to Mathoverflow http://mathoverflow.net/questions/182934/learning-roadmap-combinatorial-probability – Jyotirmoy Bhattacharya Oct 08 '14 at 16:14

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I would suggest you the following papers:

  • "Percolation and disordered systems" by G. Grimmet, 2012 reprint, an interesting paper that integrates a previous book by the same author. Here you can also find the introductory chapter of the original book, which illustrates a number of basic concepts in this context, such as bond percolation, critical phenomenon, site percolation, and so on;

  • "Percolation and the Random Cluster Model: Combinatorial and Algorithmic Problems" , another review book by Dominic Welsh that illustrates the basilar concepts of percolation theory in a very clear manner;

  • "Percolation theory and network modeling applications in soul physics" by B. Berkowitz and R.P. Ewing, 1998, a relatively old but still valuable and extensive review on the links between percolation theory and network modeling, with a particular focus on the role of these theories in explaining the randomness component of porous media behaviour;

  • "Conformal invariance of lattice models" by H. Duminil-Copin and S. Smirnov, 2012, a very high-level review on conformal invariance of the planar Ising model that provides rigorous mathematical details on several interesting topics, including FK percolation (see in particular sections 3 and 8), theory of discrete holomorphic functions, and their applications to planar statistical physics;

  • "Stochastics" by H.O. Georgii, 2008, a very comprehensive book on stochastic theories and analyses that includes well-written sections dedicated to Markov chains;

  • "Markov chains" by J. Norris, 2008, a very interesting book that deals with this topic from a strictly mathematical point of view;

  • "Reversible Markov Chains and Random Walks on Graphs" , an interesting monograph on this topic, collecting many articles written by D. Aldous and J. Allen Fill over different years (recompiled version, 2014).

Anatoly
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