  We study vortex states in a 3d random-field XY model of up to one billion
lattice spins. Starting with random spin orientations, the sample freezes into
the vortex-glass state with a stretched-exponential decay of spin correlations,
having short correlation length and a low susceptibility, compared to
vortex-free states. In a field opposite to the initial magnetization, peculiar
topological objects -- walls of spins still opposite to the field -- emerge
along the hysteresis curve. On increasing the field strength, the walls develop
cracks bounded by vortex loops. The loops then grow in size and eat the walls
away. Applications to magnets and superconductors are discussed.
