  We study the phase transition of the three-dimensional complex |psi|^4 theory
by considering the geometrically defined vortex-loop network as well as the
magnetic properties of the system in the vicinity of the critical point. Using
high-precision Monte Carlo techniques we examine an alternative formulation of
the geometrical excitations in relation to the global O(2)-symmetry breaking,
and check if both of them exhibit the same critical behavior leading to the
same critical exponents and therefore to a consistent description of the phase
transition. Different percolation observables are taken into account and
compared with each other. We find that different definitions of constructing
the vortex-loop network lead to different results in the thermodynamic limit,
and the percolation thresholds do not coincide with the thermodynamic phase
transition point.
