  The compact Abelian Higgs model is simulated on a cubic lattice where it
possesses vortex lines and pointlike magnetic monopoles as topological defects.
The focus of this high-precision Monte Carlo study is on the vortex network,
which is investigated by means of percolation observables. In the region of the
phase diagram where the Higgs and confinement phases are separated by a
first-order transition, it is shown that the vortices percolate right at the
phase boundary, and that the first-order nature of the transition is reflected
by the network. In the crossover region, where the phase boundary ceases to be
first order, the vortices are shown to still percolate. In contrast to other
observables, the percolation observables show finite-size scaling. The
exponents characterizing the critical behavior of the vortices in this region
are shown to fall in the random percolation universality class.
