  The three-dimensional XY model is studied at finite chemical potential using
complex Langevin dynamics. The validity of the approach is probed at small
chemical potential using imaginary chemical potential and continuity arguments,
and at larger chemical potential by comparison with the world line method.
While complex Langevin works for larger beta, we find that it fails for smaller
beta, in the region of the phase diagram corresponding to the disordered phase.
Diagnostic tests are developed to identify symptoms correlated with incorrect
convergence. We argue that the erroneous behaviour at smaller beta is not due
to the sign problem, but rather resembles dynamics observed in complex Langevin
simulations of simple models with complex noise.
