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.