  We present a study of galaxy mergers up to $z=10$ using the Planck Millennium
cosmological dark matter simulation and the {\tt GALFORM} semi-analytical model
of galaxy formation. Utilising the full ($800$ Mpc)$^3$ volume of the
simulation, we studied the statistics of galaxy mergers in terms of merger
rates and close pair fractions. We predict that merger rates begin to drop
rapidly for high-mass galaxies ($M_*>10^{11.3}-10^{10.5}$ $M_\odot$ for
$z=0-4$), as a result of the exponential decline in the galaxy stellar mass
function. The predicted merger rates increase and then turn over with
increasing redshift, by $z=3.5$, in disagreement with hydrodynamical
simulations and semi-empirical models. In agreement with most other models and
observations, we find that close pair fractions flatten or turn over at some
redshift (dependent on the mass selection). We conduct an extensive comparison
of close pair fractions, and highlight inconsistencies among models, but also
between different observations. We provide a fitting formula for the major
merger timescale for close galaxy pairs, in which the slope of the stellar mass
dependence is redshift dependent. This is in disagreement with previous
theoretical results that implied a constant slope. Instead we find a weak
redshift dependence only for massive galaxies ($M_*>10^{10}$ M$_\odot$): in
this case the merger timescale varies approximately as $M_*^{-0.55}$. We find
that close pair fractions and merger timescales depend on the maximum projected
separation as $r_\mathrm{max}^{1.32}$. This is in agreement with observations
of small-scale clustering of galaxies, but is at odds with the linear
dependence on projected separation that is often assumed.
