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I have vaguely heard of this connection between random matrix theory and graphs (the spectral gap of their laplacians) on compact Riemann surfaces.

  • Can someone give a pedagogic reference which helps learn this subject?

  • Also is there something called "free probability theory" which is somehow equivalent to random matrix theory?

  • Would you call the usual QFT of Hermitian matrices a random matrix theory?

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    $\begingroup$ Why has this question been downvoted!? $\endgroup$
    – user6818
    Commented Jul 21, 2014 at 6:36
  • $\begingroup$ think its borderline research level. there are large amts of refs easily located. see eg network solutions by Klarreich for a nice pop sci overview incl recent research $\endgroup$
    – vzn
    Commented Jul 23, 2014 at 4:21

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Suggest these free available resources. Most textbooks on graph theory only glance at "free probability theory".

Lectures on the Combinatorics of Free Probability (London Mathematical Society Lecture Note Series): http://www.amazon.com/Lectures-Combinatorics-Probability-Mathematical-Society/dp/0521858526

Speicher lectures: http://www.mast.queensu.ca/~speicher/papers/lectures-IHP.pdf

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  • $\begingroup$ URL : books.google.com/… $\endgroup$
    – Paul Harrington
    Commented Jul 21, 2014 at 17:24
  • $\begingroup$ But do these references build the connection with grpahs on Riemann surfaces? (and may be their Laplacians) I m not able to locate it. $\endgroup$
    – user6818
    Commented Jul 23, 2014 at 15:33

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