  For a system of N identical particles in a random pure state, there is a
threshold k_0 = k_0(N) ~ N/5 such that two subsystems of k particles each
typically share entanglement if k > k_0, and typically do not share
entanglement if k < k_0. By "random" we mean here "uniformly distributed on the
sphere of the corresponding Hilbert space." The analogous phase transition for
the positive partial transpose (PPT) property can be described even more
precisely. For example, for N qubits the two subsystems of size k are typically
in a PPT state if k < k_1 := N/4 - 1/2, and typically in a non-PPT state if k >
k_1. Since, for a given state of the entire system, the induced state of a
subsystem is given by the partial trace, the above facts can be rephrased as
properties of random induced states. An important step in the analysis depends
on identifying the asymptotic spectral density of the partial transposes of
such random induced states, a result which is interesting in its own right.
