  In discussing the phase transition of the three-dimensional complex |psi|^4
theory, we study 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 investigate if both of them exhibit
the same critical behavior leading to the same critical exponents and hence to
a consistent description of the phase transition. Different percolation
observables are taken into account and compared with each other. We find that
different connectivity definitions for 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.
