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The compact Abelian Higgs model is simulated on a cubic lattice where itpossesses 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 thephase diagram where the Higgs and confinement phases are separated by afirst-order transition, it is shown that the vortices percolate right at thephase boundary, and that the first-order nature of the transition is reflectedby the network. In the crossover region, where the phase boundary ceases to befirst order, the vortices are shown to still percolate. In contrast to otherobservables, the percolation observables show finite-size scaling. Theexponents characterizing the critical behavior of the vortices in this regionare shown to fall in the random percolation universality class.

Percolation of Vortices in the 3D Abelian Lattice Higgs Model