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.
Tom Alberts. Firas Rassoul-Agha. Mackenzie Simper. "Busemann functions and semi-infinite O’Connell–Yor polymers." Bernoulli 26 (3) 1927 - 1955, August 2020. https://doi.org/10.3150/19-BEJ1177