# Stationary Directed Polymers and Energy Solutions of the Burgers Equation

## TL;DR

This paper demonstrates that the increments of the log-partition function in a semi-discrete directed polymer model converge to energy solutions of the stochastic Burgers equation, using a novel approach that bypasses traditional transforms and spectral gap estimates.

## Contribution

It introduces a new proof technique for convergence to the stochastic Burgers equation without relying on the Cole-Hopf transform or spectral gap estimates.

## Key findings

- Convergence of log-partition function increments to energy solutions of the stochastic Burgers equation.
- Development of a second-order Boltzmann-Gibbs principle for the model.
- A proof approach that simplifies analysis by avoiding spectral gap estimates.

## Abstract

We consider the stationary O'Connell-Yor model of semi-discrete directed polymers in a Brownian environment in the intermediate disorder regime and show convergence of the increments of the log-partition function to the energy solutions of the stochastic Burgers equation. The proof does not rely on the Cole-Hopf transform and avoids the use of spectral gap estimates for the discrete model. The key technical argument is a second-order Boltzmann-Gibbs principle.