  We consider random stochastic matrices $M$ with elements given by
$M_{ij}=|U_{ij}|^2$, with $U$ being uniformly distributed on one of the
classical compact Lie groups or associated symmetric spaces. We observe
numerically that, for large dimensions, the spectral statistics of $M$,
discarding the Perron-Frobenius eigenvalue $1$, are similar to those of the
Gaussian Orthogonal ensemble for symmetric matrices and to those of the real
Ginibre ensemble for non-symmetric matrices. Using Weingarten functions, we
compute some spectral statistics that corroborate this universality. We also
establish connections with some difficult enumerative problems involving
permutations.
