  We investigate the properties of uniform doubly stochastic random matrices,
that is non-negative matrices conditioned to have their rows and columns sum to
1. The rescaled marginal distributions are shown to converge to exponential
distributions and indeed even large sub-matrices of side-length
$o(n^{1/2-\epsilon})$ behave like independent exponentials. We determine the
limiting empirical distribution of the singular values the the matrix. Finally
the mixing time of the associated Markov chains is shown to be exactly 2 with
high probability.
