  We investigate a model of random spatial permutations on two-dimensional
tori, and establish that the joint distribution of large cycles is
asymptotically given by the Poisson--Dirichlet distribution with parameter one.
The asymmetry of the tori we consider leads to a spatial bias in the
permutations, and this allows for a simple argument to deduce the existence of
mesoscopic cycles. The main challenge is to leverage this mesoscopic structure
to establish the existence and distribution of macroscopic cycles. We achieve
this by a dynamical resampling argument in conjunction with a method developed
by Schramm for the study of random transpositions on the complete graph. Our
dynamical analysis implements generic heuristics for the occurrence of the
Poisson--Dirichlet distribution in random spatial permutations, and hence may
be of more general interest.
