  We present a new N-body simulation from the MICE collaboration, the MICE
Grand Challenge (MICE-GC), containing about 70 billion dark-matter particles in
a (3 Gpc/h)^3 comoving volume. Given its large volume and fine spatial
resolution, spanning over 5 orders of magnitude in dynamic range, it allows an
accurate modeling of the growth of structure in the universe from the linear
through the highly non-linear regime of gravitational clustering. We validate
the dark-matter simulation outputs using 3D and 2D clustering statistics, and
discuss mass-resolution effects in the non-linear regime by comparing to
previous simulations and the latest numerical fits. We show that the MICE-GC
run allows for a measurement of the BAO feature with percent level accuracy and
compare it to state-of-the-art theoretical models. We also use sub-arcmin
resolution pixelized 2D maps of the dark-matter counts in the lightcone to make
tomographic analyses in real and redshift space. Our analysis shows the
simulation reproduces the Kaiser effect on large scales, whereas we find a
significant suppression of power on non-linear scales relative to the real
space clustering. We complete our validation by presenting an analysis of the
3-point correlation function in this and previous MICE simulations, finding
further evidence for mass-resolution effects. This is the first of a series of
three papers in which we present the MICE-GC simulation, along with a wide and
deep mock galaxy catalog built from it. This mock is made publicly available
through a dedicated webportal, http://cosmohub.pic.es.
