Busemann functions and semi-infinite O’Connell–Yor polymers

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.

The directed landscape

“…The single-path Busemann process and semi-infinite geodesics in Brownian LPP have been extensively studied in recent years. The state-of-the-art results are due to Seppäläinen and Sorensen [SS23b], building on [ARAS20,SS23a]. They showed that there is a random countable set Θ ⊂ [0, ∞) such that for θ ∉ Θ, the limit (8) exists for all x ∈ R and the process (θ, x) ↦ B θ (x; B) is continuous in both x and θ on R × [0, ∞) ∖ Θ.…”

Dynamic polymers: invariant measures and ordering by noise

“…Describing thermodynamic limits is a fundamental problem for Gibbs distributions, and there is a growing interest to IVPMs in random environment, especially in connection with the study of Busemann functions playing the role of global solutions of appropriate analogues of the Burgers/KPZ equations. Besides [BL19], [BL18], see [BK10], [GRASY15], [GRAS16], [ARAS20], [JRA20a], [JRA20b] studying lattice polymer models.…”