Random permutations show up in a variety of areas in
mathematics and its applications. In connection with physical
applications (e.g., the lambda transition for superfluid helium), there
is an interest in random spatial permutations -- that is, laws on
permutations that have a 'geometric bias'. There are compelling
heuristic arguments that this spatial bias has little effect on the
distribution of the largest cycles of a random spatial permutation,
provided that large cycles actually exist. I'll discuss a particular
model of random spatial permutations (directed permutations on
asymmetric tori) where these heuristics can be made precise, and large
cycles can be shown to follow the expected (Poisson-Dirichlet) law.
Based on joint work with Alan Hammond.