Higher‐order analogues of the Tracy‐Widom distribution and the Painlevé II hierarchy

We study Fredholm determinants related to a family of kernels that 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 Painlevé 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 Painlevé II hierarchy. In addition, we compute large gap asymptotics for the Fredholm determinants.