Quantum Heisenberg models and random loop representations

Abstract: 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.