  Using a Coulomb gas method, we compute analytically the probability
distribution of the Renyi entropies (a standard measure of entanglement) for a
random pure state of a large bipartite quantum system. We show that, for any
order q>1 of the Renyi entropy, there are two critical values at which the
entropy's probability distribution changes shape. These critical points
correspond to two different transitions in the corresponding charge density of
the Coulomb gas: the disappearance of an integrable singularity at the origin
and the detachement of a single-charge drop from the continuum sea of all the
other charges. These transitions respectively control the left and right tails
of the entropy's probability distribution, as verified also by Monte Carlo
numerical simulations of the Coulomb gas equilibrium dynamics.
