  Spatial random permutations were originally studied due to their connections
to Bose-Einstein condensation, but they possess many interesting properties of
their own. For random permutations of a regular lattice with periodic boundary
conditions, we prove existence of the infinite volume limit under fairly weak
assumptions. When the dimension of the lattice is two, we give numerical
evidence of a Kosterlitz-Thouless transition, and of long cycles having an
almost sure fractal dimension in the scaling limit. Finally we comment on
possible connections to Schramm-L\"owner curves.
