Unitary ensembles with a critical edge point, their multiplicative statistics and the Korteweg-de-Vries hierarchy

We study the multiplicative statistics associated to the limiting determinantal point process describing unitary random matrices with a critical edge point, where limiting density vanishes like a power 5/2. We prove that these statistics are governed by the first three equations of the KdV hierarchy, and study the asymptotic behavior of the relevant solutions.