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.