# The free boundary Schur process and applications I

## TL;DR

This paper introduces the free boundary Schur process, extending the classical Schur process by allowing boundary partitions to be arbitrary, and explores its correlation functions, pfaffian properties, and applications to percolation models and partitions.

## Contribution

It develops a comprehensive framework for the free boundary Schur process, including correlation functions, pfaffian proofs, and new applications in combinatorics and probability.

## Key findings

- Correlation functions computed via free fermion formalism.
- Proof that the process is pfaffian with one free boundary.
- Identification of non-pfaffian nature with two free boundaries and related processes.

## Abstract

We investigate the free boundary Schur process, a variant of the Schur process introduced by Okounkov and Reshetikhin, where we allow the first and the last partitions to be arbitrary (instead of empty in the original setting). The pfaffian Schur process, previously studied by several authors, is recovered when just one of the boundary partitions is left free. We compute the correlation functions of the process in all generality via the free fermion formalism, which we extend with the thorough treatment of "free boundary states". For the case of one free boundary, our approach yields a new proof that the process is pfaffian. For the case of two free boundaries, we find that the process is not pfaffian, but a closely related process is. We also study three different applications of the Schur process with one free boundary: fluctuations of symmetrized last passage percolation models, limit shapes and processes for symmetric plane partitions, and for plane overpartitions.