Does random matrix theory make any prediction for the eigenvalue distributions of compact Riemann surfaces? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T09:49:51Z http://mathoverflow.net/feeds/question/69988 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69988/does-random-matrix-theory-make-any-prediction-for-the-eigenvalue-distributions-of Does random matrix theory make any prediction for the eigenvalue distributions of compact Riemann surfaces? Marc Palm 2011-07-11T06:06:18Z 2011-07-11T15:49:52Z <p>Under RH, Montgomery has proven equidistribution results for the zeros of the Riemann Zeta function, which suggest a close connection of the distribution to certain results in Random matrix theory. Analogues have been proven for zeta function associated finite fields unconditionally.</p> <p>The Riemann zeros with imaginary part less than \$T\$ grow like \$T \log T\$. The zeros of the Selberg zeta function of a compact surface, which are connected to the eigenvalues of the Laplace Beltrami operator, grow roughly like \$T^2\$, but here the Riemann hypothesis is true except for possible zeros with imaginary part being zero.</p> <p>Nevertheless, I dare to ask, if there is something similar available for these eigenvalue?</p> http://mathoverflow.net/questions/69988/does-random-matrix-theory-make-any-prediction-for-the-eigenvalue-distributions-of/69990#69990 Answer by Alex Eskin for Does random matrix theory make any prediction for the eigenvalue distributions of compact Riemann surfaces? Alex Eskin 2011-07-11T06:38:42Z 2011-07-11T06:46:52Z <p>You can look at the following survey by Peter Sarnak:</p> <p><a href="http://www.math.princeton.edu/sarnak/Arithmetic%20Quantum%20Chaos.pdf" rel="nofollow">http://www.math.princeton.edu/sarnak/Arithmetic%20Quantum%20Chaos.pdf</a></p> <p>Basically the prediction is that the eigenvalue distribution is Poisson for arithmetic surfaces and GOE for non-arithmetic surfaces. There are some partial results supporting Poisson in the arithmetic case, in particular by Luo and Sarnak. </p> <p>This survey has a lot of really cool stuff, but is quite dated by now. For one, it does not include the solution to the quantum unique ergodicity conjecture (which among other things got Lindenstrauss the Fields medal). </p>