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Busemann functions and Gibbs measures in directed polymer models on $\mathbb{Z}^2$
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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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Cited by 2 Pith papers
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Rooted Gibbs-DLR Measures in Planar Directed Polymers
Extremal rooted Gibbs-DLR measures in planar directed polymers form a closed, totally ordered, coalescing family and generate a unique canonical Busemann process with L1 continuity.
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Geometry of geodesics through Busemann measures in directed last-passage percolation
In exponential directed last-passage percolation, the full family of semi-infinite geodesics is characterized: unique coalescing trees in all but a countable dense set of directions, exactly two trees in the exception...
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