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We discuss variational formulas for the limits of certain models of motion ina random medium: namely, the limiting time constant for last-passagepercolation and the limiting free energy for directed polymers. The results arevalid for models in arbitrary dimension, steps of the admissible paths can begeneral, the environment process is ergodic under shifts, and the potentialaccumulated along a path can depend on the environment and the next step of thepath. The variational formulas come in two types: one minimizes overgradient-like cocycles, and another one maximizes over invariant measures onthe space of environments and paths. Minimizing cocycles can be obtained fromBusemann functions when these can be proved to exist. The results areillustrated through 1+1 dimensional exactly solvable examples, periodicexamples, and polymers in weak disorder.

Variational formulas and cocycle solutions for directed polymer and percolation models