DIRECTED PATHS IN A RANDOM POTENTIAL

The properties of directed paths in random media are explored, with emphasis on the low-temperature phase. Scaling arguments, numerical simulations, and exact results are all utilized. Some of the results presented concern the existence of large-scale low-free-energy excitations in the low-temperature phase, sample-to-sample entropy variations which are much larger than the free-energy variations, and concomitant sensitivity of the optimal configuration to temperature changes, analogously to spin glasses. Nevertheless, it is argued that at fixed temperature, the possible states of a directed path in an infinite system with one end fixed are simply parametrized by its average orientation. Possibilities for the behavior in high dimensions are examined and some of the pathologies of the system on Cayley trees are discussed.