  Entanglement properties of random multipartite quantum states which are
invariant under global SU($d$) action are investigated. The random states live
in the tensor power of an irreducible representation of SU($d$). We calculate
and analyze the expectation and fluctuation of the second-order R\'enyi
entanglement measure of the random invariant and near-invariant states in high
dimension, and reveal the phenomenon of concentration of measure the random
states exhibit. We show that with high probability a random SU($d$)-invariant
state is close to being maximally entangled with respect to any bipartite cut
as the dimension of individual system goes to infinity. We also show that this
generic entanglement property of random SU(2)-invariant state is robust to
arbitrarily finite disturbation.
