  (Abridged) We present here the second half of an ESO Large Programme, which
exploits the unique combination of area and sensitivity provided in the near-IR
by the camera Hawk-I at the VLT. We have obtained - 30 observing hours with
Hawk-I in the Y-band of two high galactic latitude fields. We combined the
Y-band data with deep J and K Hawk-I observations, and with FORS1/FORS2 U, B,
V, R, I, and Z observations to select z-drop galaxies having Z - Y > 1, no
optical detection and flat Y - J and Y - K colour terms. We detect 8
high-quality candidates in the magnitude range Y = 25.5 - 26.5 that we add to
the z-drop candidates selected in two Hawk-I pointings over the GOODS-South
field. We use this full sample of 15 objects found in -161 arcmin^2 of our
survey to constrain the average physical properties and the evolution of the
number density of z ~ 7 LBGs. A stacking analysis yields a best-fit SED with
photometric redshift z= 6.85 +0.20 -0.15 and an E(B-V)=0.05 +0.15 -0.05. We
compute a binned estimate of the z ~ 7 LF and explore the effects of
photometric scatter and model uncertainties on the statistical constraints.
After accounting for the expected incompleteness through MonteCarlo
simulations, we strengthen our previous finding that a Schechter luminosity
function constant from z=6 to z=7 is ruled out at a >99% confidence level, even
including the effects of cosmic variance. For galaxies brighter than M_1500=
-19.0, we derive a luminosity density rho_UV = 1.5^{+2.1}{-0.8} x 10^25
erg/s/Hz/Mpc^3, implying a decrease by a factor 3.5 from z=6 to z=6.8. We find
that, under standard assumptions, the emission rate of ionizing photons coming
from UV bright galaxies is lower by at least a factor of two than the value
required for reionization. Finally, we exploit deep Hawk-I J and K band
observations to derive an upper limit on the number density of M1500<~ -22.0
LBGs at z-8 (Y-dropouts).
