Multiplying unitary random matrices—universality and spectral properties

In this paper, we calculate, in the large N limit, the eigenvalue density of an infinite product of random unitary matrices, each of them generated by a random Hermitian matrix. This is equivalent to solving the unitary diffusion generated by a Hamiltonian random in time. We find that the average eigenvalue density is universal and depends only on the second moment of the generator of the stochastic evolution. We find indications of critical behaviour (eigenvalue spacing scaling like 1/N^3/4) close to thgr = pgr for a specific critical evolution time t_c.