Symmetrized random permutations

Selecting N random points in a unit square corresponds to selecting a random permutation. Placing symmetry restrictions on the points, we obtain special kinds of permutations : involutions, signed permutations and signed involutions. We are interested in the statistics of the length (in numbers of points) of the longest up/right path in each symmetry type as the number of points increases to infinity. The limiting distribution functions are expressed in terms of a Painlevé II equation. In addition to the Tracy-Widom distributions of random matrix theory, we also obtain two new classes of distribution functions interpolating between the GOE and GSE, and between the GUE and GOE Tracy-Widom distribution functions. Applications to random vicious walks and site percolation are also discussed