# Busemann functions and semi-infinite O'Connell-Yor polymers

## TL;DR

This paper establishes the existence of infinite length limits for the O'Connell-Yor polymer and Brownian last passage percolation, using Busemann functions and Markovian properties to prove a law of large numbers.

## Contribution

It introduces a method to prove the existence of Busemann functions for these models, linking finite and infinite length behaviors.

## Key findings

- Existence of infinite length limits for O'Connell-Yor polymer.
- Existence of infinite length limits in Brownian last passage percolation.
- Busemann functions serve as key tools in analyzing these limits.

## Abstract

We prove that given any fixed asymptotic velocity, the finite length O'Connell-Yor polymer has an infinite length limit satisfying the law of large numbers with this velocity. By a Markovian property of the quenched polymer this reduces to showing the existence of Busemann functions: almost sure limits of ratios of random point-to-point partition functions. The key ingredients are the Burke property of the O'Connell-Yor polymer and a comparison lemma for the ratios of partition functions. We also show the existence of infinite length limits in the Brownian last passage percolation model.