Higher order analogues of the Tracy-Widom distribution and the Painleve II hierarchy

We study Fredholm determinants related to a family of kernels which describe the edge eigenvalue behavior in unitary random matrix models with critical edge points. The kernels are natural higher order analogues of the Airy kernel and are built out of functions associated with the Painlev\'e I hierarchy. The Fredholm determinants related to those kernels are higher order generalizations of the Tracy-Widom distribution. We give an explicit expression for the determinants in terms of a distinguished smooth solution to the Painlev\'e II hierarchy. In addition we compute large gap asymptotics for the Fredholm determinants.