Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
4  
There are an awful lot of Google hits for "L-functions and random matrices." –  Qiaochu Yuan Nov 16 '09 at 23:31
    
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. –  Thomas Riepe Nov 16 '09 at 23:53

5 Answers 5

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

share|improve this answer
    
I know that and it is surveyed in the article mentioned above, but I am looking for other texts. –  Thomas Riepe Nov 17 '09 at 9:33

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

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

share|improve this answer

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

share|improve this answer

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.

share|improve this answer

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

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.