  We study a class of quantum spin systems that includes the $S=\tfrac12$
Heisenberg and XY-models, and prove that two-point correlations exhibit
exponential decay in the presence of a transverse magnetic field. The field is
not necessarily constant, it may be random, and it points in the same
direction. Our proof is entirely probabilistic and it relies on a
random-loop-representation of the correlation functions, on stochastic
domination, and on first-passage percolation.
