  Auxiliary field Quantum Monte Carlo simulations of interacting fermions
require sampling over a Hubbard-Stratonovich field $h$ introduced to decouple
the interactions. The weight for a given configuration involves the products of
the determinant of matrices $M_\sigma(h)$, where $\sigma$ labels the species,
and hence is typically not positive definite. Indeed, the average sign $\langle
{\cal S} \rangle$ of the determinants goes to zero exponentially with
increasing spatial size and decreasing temperature for most Hamiltonians of
interest. This statement, however, does not explicitly separate two possible
origins for the vanishing of $\langle {\cal S} \rangle$. Does $\langle {\cal S}
\rangle \rightarrow 0$ because {\it randomly} chosen field configurations have
${\rm det}\big(M(h)\big) < 0$, or does the `sign problem' arise because the
specific subset of configurations chosen by the weighting function have a
greater preponderance of negative values? In the latter case, the process of
weighting the configurations with $|{\rm det}\big(M(h)\big)|$ might steer the
simulation to a region of configuration space of $h$ where positive and
negative determinants are equally likely, even though randomly chosen $h$ would
preferentially have determinants with a single dominant sign. In this paper we
address the relative importance of these two mechanisms for the vanishing of
$\langle {\cal S} \rangle$ in quantum simulations.
