# Does random matrix theory make any prediction for the eigenvalue distributions of compact Riemann surfaces?

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.

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.

Nevertheless, I dare to ask, if there is something similar available for these eigenvalue?

-

## 1 Answer

You can look at the following survey by Peter Sarnak:

http://www.math.princeton.edu/sarnak/Arithmetic%20Quantum%20Chaos.pdf

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.

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).

-
Thanks a lot for the short summary and the reference. – Marc Palm Jul 11 '11 at 9:49