Stochastic bias in multi-dimensional excursion set approaches

Understand

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.