# Random Matrix Theory

Could the distribution of the prime numbers be related to a physical system?
It seems that the distribution of eigenvalues in quantum chaotic systems obeys the same statistics as eigenvalues of random matrices. e.g. http://www.researchgate.net/publication/51969774_Generalized_random_matrix_conjecture_for_chaotic_systems It has also been shown that the distribution of the critical zeros of the Riemann zeta-function can be related to the distribution of eigenvalues of certain random matrices: http://www.researchgate.net/publication/51940315_Random_matrices_and_Riemann_hypothesis So this would seem to to suggest a statistical relationship between the zeros of the zeta function and the energy levels some quantum chaotical system. Can any relationship be drawn between the placement of the prime numbers and a dynamical system? Any other implications of this connection?
Marcel Novaes · Universidade Federal de São Carlos
Berry and Keating have conjectured that there should exist a chaotic system such that the periods of the periodic orbits are the logarithms of the prime numbers. The quantization of this system would have the Riemann zeroes as eigtenvalues.
Thank James for your p.m. answer. Let me share it, and comment it in the context of traffics. __________ By James Mingo: "First regarding your post: according to Benoit's formula $\mathrm{E}(u_{i_1j_1} \cdots u_{i_nj_n} \overline{u_{i_{1}'j_{1}'}} \cdots \overline{u_{i_{n}'j_{n}'}})$ = \sum_{\pi, \sigma \in S_n} \delta_{i_1 i_{\pi(1)}'} \cdots \delta_{i_n i_{\pi(n)}'} \delta_{j_1 {j}_{\sigma(1)}'} \cdots \delta_{j_n j_{\sigma(n)}'} \mathrm{Wg}(\pi^{-1}\sigma) \] We have (because there no complex conjugates) E(u_{ij}u_{ik}) = 0 as well as E(u_{ij}) = 0. Thus the covariance is 0. If we replace U by O (the orthogonal group) we still get E(o_{ij}o_{ik}) = 0 for k and k not equal. E(o_{ij})^2) = 1/N where N is the size of the matrix." ___________ I have then realized a simple interesting fact, a simple way to observe the difference between the limiting distribution of permutation matrices and Haar matrices in the classical compact groups. For any n.c.v.r. "a" in a space of traffics, thanks to certain operad construction I'll present soon in my revised version of [1], I can define the operator "deg(a)", which correspond, for a matrix "A = (A_{i,j})_{i,j=1..N}" to the diagonal matrix "deg(A) = diag( \sum_{j=1..N} A_{ij} )_{i=1..N}" If "u" is the limit in distribution of traffics of a uniform permutation matrix (or any permutation matrix with small density of cycle of given size), then "deg(u)" is the identity "\mbf 1" in the underlying *-probability space. Nevertheless, if "u" is the limit of a Haar matrix on the unitary group, then "deg(u)" has the distribution of "\theta \mbf 1", where "\theta" is a complex gaussian random variable.