Universal edge scaling in random partitions

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 Schrodinger-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.

Automated summary

In this study, the universal edge scaling limit of random partitions with the Schur measure was established. The asymptotic behavior of the wave function, based on a Schrodinger-type differential equation, was explored. It was shown that in the edge scaling limit, the wave function generally approaches the Airy function and its higher-order analogs. Additionally, a higher-order Airy kernel and the Tracy-Widom distribution were constructed from the wave function in the scaling limit, with implications for multicritical phase transitions in large-size matrix models. The limit shape of random partitions was also discussed through semi-classical analysis of the wave function.