  On large scales galaxies and their halos are usually assumed to trace the
dark matter with a constant bias and dark matter is assumed to trace the linear
density field. We test these assumption using several large N-body simulations
with 384^3-1024^3 particles and box sizes between 100-1000h/Mpc, which can both
resolve the small galactic size halos and sample the large scale fluctuations.
We explore the average halo bias relation as a function of halo mass and show
that existing fitting formulae overestimate the halo bias by up to 20% in the
regime just below the nonlinear mass. We propose a new expression that fits our
simulations well. We find that the halo bias is nearly constant, b~0.65-0.7,
for masses below one tenth of the nonlinear mass. We explore next the relation
between the initial and final dark matter in individual Fourier modes and show
that there are significant fluctuations in their ratio, ranging from 10% rms at
k~0.03h/Mpc to 50% rms at k~0.1h/Mpc. We argue that these large fluctuations
are caused by perturbative effects beyond the linear theory, which are
dominated by long wavelength modes with large random fluctuations. Similar or
larger fluctuations exist between halos and dark matter and between halos of
different mass. While these fluctuations are small compared to the sampling
variance, they are significant for attempts to determine the bias by relating
directly the maps of galaxies and dark matter or the maps of different galaxy
populations, which would otherwise be immune to sampling variance.
