L-functions and random matrices - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:22:43Z http://mathoverflow.net/feeds/question/5750 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/5750/l-functions-and-random-matrices L-functions and random matrices Thomas Riepe 2009-11-16T23:24:07Z 2011-01-03T20:59:21Z <p>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)? </p> <p>A <a href="http://math.arizona.edu/~ulmer/research/NS.pdf" rel="nofollow" title="pdf">very nice survey</a> 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 Weilâ€™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.</p> http://mathoverflow.net/questions/5750/l-functions-and-random-matrices/5783#5783 Answer by Alon Amit for L-functions and random matrices Alon Amit 2009-11-17T05:42:55Z 2009-11-17T05:42:55Z <p>The standard reference is (or at least used to be) <a href="http://www.amazon.com/exec/obidos/ASIN/0821810170" rel="nofollow">Katz and Sarnak</a>. </p> http://mathoverflow.net/questions/5750/l-functions-and-random-matrices/7442#7442 Answer by PeterR for L-functions and random matrices PeterR 2009-12-01T15:07:27Z 2009-12-01T15:07:27Z <p>Have a look at J. Brian Conrey's "L-functions and Random Matrices" at</p> <p><a href="http://arxiv.org/PS_cache/math/pdf/0005/0005300v1.pdf" rel="nofollow">http://arxiv.org/PS_cache/math/pdf/0005/0005300v1.pdf</a></p> http://mathoverflow.net/questions/5750/l-functions-and-random-matrices/38101#38101 Answer by Micah Milinovich for L-functions and random matrices Micah Milinovich 2010-09-08T21:13:40Z 2010-09-08T21:13:40Z <p>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:</p> <p><a href="http://www.amazon.com/gp/product/0521620589/qid=1141005450/sr=1-2/ref=sr_1_2?s=books&amp;v=glance&amp;n=283155" rel="nofollow">http://www.amazon.com/gp/product/0521620589/qid=1141005450/sr=1-2/ref=sr_1_2?s=books&amp;v=glance&amp;n=283155</a></p> <p>Similarly, here is one for the connection between the ranks of elliptic curves and random matrix theory:</p> <p><a href="http://www.amazon.com/Elliptic-Curves-Mathematical-Society-Lecture/dp/0521699649/ref=sr_1_2?s=books&amp;ie=UTF8&amp;qid=1283980329&amp;sr=1-2" rel="nofollow">http://www.amazon.com/Elliptic-Curves-Mathematical-Society-Lecture/dp/0521699649/ref=sr_1_2?s=books&amp;ie=UTF8&amp;qid=1283980329&amp;sr=1-2</a></p> http://mathoverflow.net/questions/5750/l-functions-and-random-matrices/39352#39352 Answer by Alex R. for L-functions and random matrices Alex R. 2010-09-20T01:03:05Z 2010-09-20T01:03:05Z <p>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. </p> http://mathoverflow.net/questions/5750/l-functions-and-random-matrices/51064#51064 Answer by Stopple for L-functions and random matrices Stopple 2011-01-03T20:59:21Z 2011-01-03T20:59:21Z <p>Another expository work, by Firk and Miller (same Miller as above) is "Nuclei, Primes and the Random Matrix Connection"</p> <p><a href="http://arxiv.org/abs/0909.4914" rel="nofollow">http://arxiv.org/abs/0909.4914</a></p>