  We compute analytically the statistics of the Renyi and von Neumann entropies
(standard measures of entanglement), for a random pure state in a large
bipartite quantum system. The full probability distribution is computed by
first mapping the problem to a random matrix model and then using a Coulomb gas
method. We identify three different regimes in the entropy distribution, which
correspond to two phase transitions in the associated Coulomb gas. The two
critical points correspond to sudden changes in the shape of the Coulomb charge
density: the appearance of an integrable singularity at the origin for the
first critical point, and the detachement of the rightmost charge (largest
eigenvalue) from the sea of the other charges at the second critical point.
Analytical results are verified by Monte Carlo numerical simulations. A short
account of some of these results appeared recently in Phys. Rev. Lett. {\bf
104}, 110501 (2010).
