  The convex set of quantum states of a composite $K \times K$ system with
positive partial transpose is analysed. A version of the hit and run algorithm
is used to generate a sequence of random points covering this set uniformly and
an estimation for the convergence speed of the algorithm is derived. For $K\ge
3$ this algorithm works faster than sampling over the entire set of states and
verifying whether the partial transpose is positive. The level density of the
PPT states is shown to differ from the Marchenko-Pastur distribution, supported
in [0,4] and corresponding asymptotically to the entire set of quantum states.
Based on the shifted semi--circle law, describing asymptotic level density of
partially transposed states, and on the level density for the Gaussian unitary
ensemble with constraints for the spectrum we find an explicit form of the
probability distribution supported in [0,3], which describes well the level
density obtained numerically for PPT states.
