Stochastic bias in multi-dimensional excursion set approaches
  We describe a simple fully analytic model of the excursion set approach
associated with two Gaussian random walks: the first walk represents the
initial overdensity around a protohalo, and the second is a crude way of
allowing for other factors which might influence halo formation. This model is
richer than that based on a single walk, because it yields a distribution of
heights at first crossing. We provide explicit expressions for the
unconditional first crossing distribution which is usually used to model the
halo mass function, the progenitor distributions, and the conditional
distributions from which correlations with environment are usually estimated.
These latter exhibit perhaps the simplest form of what is often called nonlocal
bias, and which we prefer to call stochastic bias, since the new bias effects
arise from `hidden-variables' other than density, but these may still be
defined locally. We provide explicit expressions for these new bias factors. We
also provide formulae for the distribution of heights at first crossing in the
unconditional and conditional cases. In contrast to the first crossing
distribution, these are exact, even for moving barriers, and for walks with
correlated steps. The conditional distributions yield predictions for the
distribution of halo concentrations at fixed mass and formation redshift. They
also exhibit assembly bias like effects, even when the steps in the walks
themselves are uncorrelated. Finally, we show how the predictions are modified
if we add the requirement that halos form around peaks: these depend on whether
the peaks constraint is applied to a combination of the overdensity and the
other variable, or to the overdensity alone. Our results demonstrate the power
of requiring models to reproduce not just halo counts but the distribution of
overdensities at fixed protohalo mass as well.
