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

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.

49 pages, 2 figures. Minor typos corrected. A shorter version will appear in the Annals of Probability

References in corpus (6)

• Limiting shape for directed percolation models
• Competition interfaces and second class particles
• Joint distribution of Busemann functions in the exactly solvable corner growth model
• Smooth invariant densities for random switching on the torus
• Busemann functions and Gibbs measures in directed polymer models on ${\mathbb{Z}}^2$
• Fluctuation Bounds For Interface Free Energies in Spin Glasses

Cited by in corpus (7)

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• Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique
• Uniqueness and ergodicity of stationary directed polymers on ${\mathbb{Z}}^2$
• Dynamic polymers: invariant measures and ordering by noise
• Invariant measures for stochastic conservation laws on the line