  We provide numerical evidence that the perturbative spectrum of anomalous
dimensions in maximally supersymmetric SU(N) Yang-Mills theory is chaotic at
finite values of N. We calculate the probability distribution of one-loop level
spacings for subsectors of the theory and show that for large N it is given by
the Poisson distribution of integrable models, while at finite values it is the
Wigner-Dyson distribution of the Gaussian orthogonal ensemble random matrix
theory. We extend these results to two-loop order and to a one-parameter family
of deformations. We further study the spectral rigidity for these models and
show that it is also well described by random matrix theory. Finally we
demonstrate that the finite-N eigenvectors possess properties of chaotic
states.
