  We review random loop representations for the spin-1/2 quantum Heisenberg
models, that are due to Toth (ferromagnet) and Aizenman-Nachtergaele
(antiferromagnet). These representations can be extended to models that
interpolate between the two Heisenberg models, such as the quantum XY model. We
discuss the relations between long-range order of the quantum spins and the
size of the loops. Finally, we describe conjectures about the joint
distribution of the lengths of macroscopic loops, and of symmetry breaking.
