Bott, Raoul (ed.) et al., Current developments in mathematics, 1997. Papers from the conference held in Cambridge, MA, USA, 1997. Boston, MA: International Press. 127-144, 145-159 (1999).
These are survey lectures on a subject which has been given much attention in the physics and number theory literature. Many but not all of the results surveyed are experimental. They concern the statistics of spacings of spectra of operators such as the Laplacian on a manifold as well as the spacings of zeros of many kinds of zeta functions. This subject originates with work of Wigner in the 1950’s who suggested that one should study the spectrum of a random Hermitian matrix as a good approximation to the energy levels of a quantum mechanical system. One arranges the eigenvalues in increasing order and then looks at the differences of adjacent eigenvalues (normalized to have mean spacing 1).
The basic conjecture of the subject says that if the system is completely integrable then the local spacing statistics should be Poisson in the large energy level limit. If the system is chaotic then the local spacing statistics are what is called GOE (Gaussian orthogonal ensemble) if the system has time reversal symmetry and GUE (Gaussian unitary ensemble) otherwise. Here Poisson means the histograms look like , while GOE histograms look like . The actual distributions for GUE and GOE were found by Gaudin and Mehta. Sarnak notes that the distributions should be viewed as associated with symmetric spaces. Here GOE is associated to , for example. All of the infinite families of irreducible symmetric spaces of Cartan should actually be considered.
One of the examples for completely integrable Hamiltonians is the case of particles in a box – meaning geodesic motion on a flat 2 dimensional torus , where is a lattice. Here the spectrum consists of numbers for in the dual lattice. Sarnak has shown that for almost all lattices the pair correlation of the spectrum is Poisson. In the chaotic case there are no examples with explicitly computable spectra. The only rigorous tools are the trace formulas of Selberg, Gutzwiller, ... Numerical experiments lead one to expect that for arithmetic discrete groups acting on the Poincaré upper half plane the level spacings are Poisson, while for non-arithmetic the level spacings appear to be GOE.
In Lecture 2 the focus of attention becomes zeros of zeta-functions such as Riemann’s instead of spectra. Again experimental data such as that of Odlyzko are considered. Of course Selberg’s zeta-function connects the two subjects. Sarnak uses the term “Montgomery-Odlyzko law” to mean that the high zeros of any -function corresponding to a cuspidal automorphic form on a general linear group should satisfy the GUE local spacing statistics.
New evidence for this law comes from the results of N. Katz and P. Sarnak [Random matrices, Frobenius eigenvalues and monodromy, Am. Math. Soc. (1999)] on the function field analogue of the Riemann zeta function. They proved that the zeta functions of almost all curves over a finite field have GUE level spacing as the size of the field and the genus go to infinity.