  In this paper, we investigate various ways of defining the initial source
eccentricity using the Monte Carlo Glauber (MCG) approach. In particular, we
examine the participant eccentricity, which quantifies the eccentricity of the
initial source shape by the major axes of the ellipse formed by the interaction
points of the participating nucleons. We show that reasonable variation of the
density parameters in the Glauber calculation, as well as variations in how
matter production is modeled, do not significantly modify the already
established behavior of the participant eccentricity as a function of collision
centrality. Focusing on event-by-event fluctuations and correlations of the
distributions of participating nucleons we demonstrate that, depending on the
achieved event-plane resolution, fluctuations in the elliptic flow magnitude
$v_2$ lead to most measurements being sensitive to the root-mean-square, rather
than the mean of the $v_2$ distribution. Neglecting correlations among
participants, we derive analytical expressions for the participant eccentricity
cumulants as a function of the number of participating nucleons,
$\Npart$,keeping non-negligible contributions up to $\ordof{1/\Npart^3}$. We
find that the derived expressions yield the same results as obtained from
mixed-event MCG calculations which remove the correlations stemming from the
nuclear collision process. Most importantly, we conclude from the comparison
with MCG calculations that the fourth order participant eccentricity cumulant
does not approach the spatial anisotropy obtained assuming a smooth nuclear
matter distribution. In particular, for the Cu+Cu system, these quantities
deviate from each other by almost a factor of two over a wide range in
centrality.
