  The entanglement properties of random pure states are relevant to a variety
of problems ranging from chaotic quantum dynamics to black hole physics. The
averaged bipartite entanglement entropy of such states admits a volume law and
upon increasing the subregion size follows the Page curve. In this paper, we
generalize this setup to random mixed states by coupling the system to a bath
and use the partial transpose to study their entanglement properties. We
develop a diagrammatic method to incorporate partial transpose within random
matrix theory and formulate a perturbation theory in $1/L$, the inverse of the
Hilbert space dimension. We compute several quantities including the spectral
density of partial transpose (or entanglement negativity spectrum), two-point
correlator of eigenvalues, and the logarithmic negativity. As long as the bath
is smaller than the system, we find that upon sweeping the subregion size, the
logarithmic negativity shows an initial increase and a final decrease similar
to the Page curve, while it admits a plateau in the intermediate regime where
the logarithmic negativity only depends on the size of the system and of the
bath but not on how the system is partitioned. This intermediate phase has no
analog in random pure states, and is separated from the two other regimes by a
critical point. We further show that when the bath is larger than the system by
at least two extra qubits the logarithmic negativity is identically zero which
implies that there is no distillable entanglement. Using the diagrammatic
approach, we provide a simple derivation of the semi-circle law of the
entanglement negativity spectrum in the latter two regimes. We show that
despite the appearance of a semicircle distribution, reminiscent of Gaussian
unitary ensemble (GUE), the higher order corrections to the negativity spectrum
and two-point correlator deviate from those of GUE.
