Universal edge scaling in random partitions

Abstract

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 Schrödinger-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 scaling 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.