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We establish the existence of generalized Busemann functions and Gibbs-Dobrushin-Landford-Ruelle measures for a general class of lattice random walks in random potentials with finitely many admissible steps. This class encompasses directed polymers in random environments, first- and last-passage percolation, and elliptic random walks in both static and dynamic random environments in all dimensions and with minimal assumptions on the random potential. 

We give an explicit description of a family of jointly invariant measures of the KPZ equation singled out by asymptotic slope conditions. These are couplings of Brownian motions with drift, and can be extended to a cadlag process indexed by all real drift parameters. We name this process the KPZ horizon (KPZH). As a corollary, we resolve a recent conjecture by showing the existence of a random, countably infinite dense set of drift values at which the Busemann process of the KPZ equation is discontinuous. This signals instability, and shows the failure of the one force–one solution principle and the existence of at least two extremal semi-infinite polymer measures in the exceptional directions. The low-temperature limit of the KPZH is the stationary horizon (SH), the unique jointly invariant measure of the KPZ fixed point under the same slope conditions. The high-temperature limit of the KPZH is a coupling of Brownian motions that differ by linear shifts, which is jointly invariant under the Edwards–Wilkinson fixed point.