  We investigate the asymptotic behavior of the empirical eigenvalues
distribution of the partial transpose of a random quantum state. The limiting
distribution was previously investigated via Wishart random matrices indirectly
(by approximating the matrix of trace 1 by the Wishart matrix of random trace)
and shown to be the semicircular distribution or the free difference of two
free Poisson distributions, depending on how dimensions of the concerned spaces
grow. Our use of Wishart matrices gives exact combinatorial formulas for the
moments of the partial transpose of the random state. We find three natural
asymptotic regimes in terms of geodesics on the permutation groups. Two of them
correspond to the above two cases; the third one turns out to be a new matrix
model for the meander polynomials. Moreover, we prove the convergence to the
semicircular distribution together with its extreme eigenvalues under weaker
assumptions, and show large deviation bound for the latter.
