  The excursion set approach provides a framework for predicting how the
abundance of dark matter halos depends on the initial conditions. A key
ingredient of this formalism comes from the physics of halo formation: the
specification of a critical overdensity threshold (barrier) which protohalos
must exceed if they are to form bound virialized halos at a later time. Another
ingredient is statistical, as it requires the specification of the appropriate
statistical ensemble over which to average when making predictions. The
excursion set approach explicitly averages over all initial positions, thus
implicitly assuming that the appropriate ensemble is that associated with
randomly chosen positions in space, rather than special positions such as peaks
of the initial density field. Since halos are known to collapse around special
positions, it is not clear that the physical and statistical assumptions which
underlie the excursion set approach are self-consistent. We argue that they are
at least for low mass halos, and illustrate by comparing our excursion set
predictions with numerical data from the DEUS simulations.
