  We consider random bipartite quantum states obtained by tracing out one
subsystem from a random, uniformly distributed, tripartite pure quantum state.
We compute thresholds for the dimension of the system being traced out, so that
the resulting bipartite quantum state satisfies the reduction criterion in
different asymptotic regimes. We consider as well the basis-independent version
of the reduction criterion (the absolute reduction criterion), computing
thresholds for the corresponding eigenvalue sets. We do the same for other sets
relevant in the study of absolute separability, using techniques from random
matrix theory. Finally, we gather and compare the known values for the
thresholds corresponding to different entanglement criteria, and conclude with
a list of open questions.
