  We establish the universal edge scaling limit of random partitions with the
infinite-parameter distribution called the Schur measure. We explore the
asymptotic behavior of the wave function, which is a building block of the
corresponding kernel, based on the Schr{\"o}dinger-type differential equation.
We show that the wave function is in general asymptotic to the Airy function
and its higher-order analogs in the edge scaling limit. We construct the
corresponding higher-order Airy kernel and the Tracy-Widom distribution from
the wave function in the scalins limit, and discuss its implication to the
multicritical phase transition in the large size matrix model. We also discuss
the limit shape of random partitions through the semi-classical analysis of the
wave function.
