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I am curious about the connection between properties of L-functions and random matrices, and about (if existent) function field versions of that. Do you know a survey or an other article where one could get an idea of those themes and possibly related issues (e.g. which of the many sorts of L-functions are related to random matrices)?

A very nice survey on the function field case by Douglas Ulmer: "The goal of this survey is to give some insight into how well-distributed sets of matrices in classical groups arise from families of L-functions in the context of the middle column of Weil’s trilingual inscription, namely function fields of curves over finite fields. The exposition is informal and no proofs are given; rather, our aim is illustrate what is true by considering key examples." There are several other very interesting articles on his website, BTW.

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    $\begingroup$ There are an awful lot of Google hits for "L-functions and random matrices." $\endgroup$
    – Qiaochu Yuan
    Commented Nov 16, 2009 at 23:31
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    $\begingroup$ Yes, too much for a lazy me to browse them all ;) A lecture last year by Katz “Simple things we don’t know” probably surveyed that theme, but no text of it exists. I wonder what on Gauss' list of "simple things we don't know" would have been. $\endgroup$
    – Thomas Riepe
    Commented Nov 16, 2009 at 23:53

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Have a look at J. Brian Conrey's "L-functions and Random Matrices" at

http://arxiv.org/PS_cache/math/pdf/0005/0005300v1.pdf

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Here is a published proceedings to a short school held at the Newton Institute in Cambridge about the connection between random matrix theory and number theory:

http://www.amazon.com/gp/product/0521620589/qid=1141005450/sr=1-2/ref=sr_1_2?s=books&v=glance&n=283155

Similarly, here is one for the connection between the ranks of elliptic curves and random matrix theory:

http://www.amazon.com/Elliptic-Curves-Mathematical-Society-Lecture/dp/0521699649/ref=sr_1_2?s=books&ie=UTF8&qid=1283980329&sr=1-2

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The standard reference is (or at least used to be) Katz and Sarnak.

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  • $\begingroup$ I know that and it is surveyed in the article mentioned above, but I am looking for other texts. $\endgroup$
    – Thomas Riepe
    Commented Nov 17, 2009 at 9:33
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If you want something more on the expository side, "An Invitation to Modern Number Theory" by Miller and Takloo-Bighash builds up both L-functions and random matrices from the ground up, later connecting the two.

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Another expository work, by Firk and Miller (same Miller as above) is "Nuclei, Primes and the Random Matrix Connection"

http://arxiv.org/abs/0909.4914

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