title: Random loop representations for quantum spin systems

abstract:
Certain quantum spin systems have fascinating random loop
representations that allow to express the quantum correlations
in a probabilistic way. I will describe an extension of the
representations of Toth and Aizenman-Nachtergaele for the
ferromagnetic and antiferromagnetic Heisenberg models. The
representation allows to prove the existence of a phase
transition with magnetic long-range order in a quantum spin 1
system. It also allows to formulate conjectures about the
nature of the phase transition, which is still not understood.