We consider two models of random loops where we prove breaking of
translational symmetry. The first is a mirror model, where the loops
are formed by light rays bouncing in a labyrinth of randomly oriented
mirrors. The second is a probabilistic representation of a quantum
spin chain, and can be obtained as a limit of the first, for
inhomogeneous mirror weights. In the terminology of quantum spins,
this symmetry-breaking is called “dimerisation”. Based on joint works
with K. Ryan as well as with P. Muehlbacher, B. Nachtergaele and D.
Ueltschi.