Busemann functions and Gibbs measures in directed polymer models on $\mathbb{Z}^2$

Christopher Janjigian; Firas Rassoul-Agha

doi:10.1214/19-aop1375

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arXiv · 1810.03580

Busemann functions and Gibbs measures in directed polymer models on $\mathbb{Z}^2$

Christopher Janjigian·Firas Rassoul-Agha

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Abstract

We consider random walk in a space-time random potential, also known as directed random polymer measures, on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices. We construct covariant cocycles and use them to prove new results on existence, uniqueness/non-uniqueness, and asymptotic directions of semi-infinite polymer measures (solutions to the Dobrushin-Lanford-Ruelle equations). We also prove non-existence of covariant or deterministically directed bi-infinite polymer measures. Along the way, we prove almost sure existence of Busemann function limits in directions where the limiting free energy has some regularity.

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BibTeX RIS

Christopher Janjigian, Firas Rassoul-Agha. 2018-10-08. Busemann functions and Gibbs measures in directed polymer models on $\mathbb{Z}^2$. https://doi.org/10.1214/19-aop1375

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