Correlation function of the Schur process with a fixed final partition

TL;DR

This paper generalizes the Schur process to include a fixed final partition, deriving a correlation kernel and analyzing its edge scaling limit, which transitions from the extended Airy kernel under certain scalings.

Contribution

It introduces a new generalized Schur process with a fixed final partition and derives its correlation kernel, including edge scaling limits.

Findings

Derived a double integral representation of the correlation kernel.

Analyzed the edge scaling limit using saddle point analysis.

Showed the transition of the limiting kernel from the extended Airy kernel.

Abstract

We consider a generalization of the Schur process in which a partition evolves from the empty partition into an arbitrary fixed final partition. We obtain a double integral representation of the correlation kernel. For a special final partition with only one row, the edge scaling limit is also discussed by the use of the saddle point analysis. If we appropriately scale the length of the row, the limiting correlation kernel changes from the extended Airy kernel.