# Free energy distribution of the stationary O'Connell-Yor directed random polymer model

## TL;DR

This paper derives the free energy distribution for the stationary O'Connell-Yor directed polymer model, revealing the universal KPZ stationary distribution and connecting it to the stationary KPZ equation.

## Contribution

It provides a Fredholm determinant formula for the free energy distribution and links the model to the KPZ universality class in the stationary regime.

## Key findings

- Universal KPZ stationary distribution appears in the long time limit
- Derived explicit Fredholm determinant formula for free energy distribution
- Connections established with the stationary KPZ equation

## Abstract

We study the semi-discrete directed polymer model introduced by O'Connell-Yor in its stationary regime, based on our previous work on the stationary $q$-totally asymmetric simple exclusion process ($q$-TASEP) using a two-sided $q$-Whittaker process. We give a formula for the free energy distribution of the polymer model in terms of Fredholm determinant and show that the universal KPZ stationary distribution appears in the long time limit. We also consider the limit to the stationary KPZ equation and discuss the connections with previously found formulas.