  The entanglement between two arbitrary subsystems of random pure states is
studied via properties of the density matrix's partial transpose,
$\rho_{12}^{T_2}$. The density of states of $\rho_{12}^{T_2}$ is close to the
semicircle law when both subsystems have dimensions which are not too small and
are of the same order. A simple random matrix model for the partial transpose
is found to capture the entanglement properties well, including a transition
across a critical dimension. Log-negativity is used to quantify entanglement
between subsystems and analytic formulas for this are derived based on the
simple model. The skewness of the eigenvalue density of $\rho_{12}^{T_2}$ is
derived analytically, using the average of the third moment over the ensemble
of random pure states. The third moment after partial transpose is also shown
to be related to a generalization of the Kempe invariant. The smallest
eigenvalue after partial transpose is found to follow the extreme value
statistics of random matrices, namely the Tracy-Widom distribution. This
distribution, with relevant parameters obtained from the model, is found to be
useful in calculating the fraction of entangled states at critical dimensions.
These results are tested in a quantum dynamical system of three coupled
standard maps, where one finds that if the parameters represent a strongly
chaotic system, the results are close to those of random states, although there
are some systematic deviations at critical dimensions.
