Universal edge scaling in random partitions

TL;DR

This paper demonstrates the universal edge scaling limit of random partitions under the Schur measure, revealing asymptotic behaviors related to Airy functions and their implications for matrix models and phase transitions.

Contribution

It introduces the higher-order Airy kernel and Tracy-Widom distribution for the Schur measure, extending understanding of edge scaling limits in random partitions.

Findings

Wave function asymptotic to Airy and higher-order Airy functions.

Construction of higher-order Airy kernel and Tracy-Widom distribution.

Implications for multicritical phase transitions in matrix models.

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 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.