  We use the Millennium Simulation (MS) to study the statistics of LCDM halo
concentrations at z = 0. Our results confirm that the average halo
concentration declines monotonically with mass; a power-law fits well the
concentration-mass relation for over 3 decades in mass, up to the most massive
objects to form in a LCDM universe (~ 10^15 h^-1 Msol). This is in clear
disagreement with the predictions of the model proposed by Bullock et al. for
these rare objects, and agrees better with the original predictions of Navarro,
Frenk, & White. The large volume surveyed, together with the unprecedented
numerical resolution of the MS, allow us to estimate with confidence the
distribution of concentrations and, consequently, the abundance of systems with
unusual properties. About one in a hundred cluster haloes (M200 >~ 3x10^14 h^-1
Msol) have concentrations exceeding c200 = 7.5, a result that may be used to
interpret the likelihood of unusually strong massive gravitational lenses, such
as Abell 1689, in the LCDM cosmogony. A similar fraction (1 in 100) of
galaxy-sized haloes (M200 ~ 10^12 h^-1 Msol) have c200 < 4.5, an important
constraint on models that attempt to reconcile the rotation curves of low
surface-brightness galaxies by appealing to haloes of unexpectedly low
concentration. We find that halo concentrations are independent of spin once
haloes manifestly out of equilibrium are removed from the sample. Compared to
their relaxed brethren, the concentrations of out-of-equilibrium haloes tend to
be lower and to have more scatter, while their spins tend to be higher...
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