Ground-state energy anisotropy for directed polymers in random media

The orientation dependence of the line energy of zero-temperature directed polymers is studied analytically and numerically for the two-dimensional square lattice with site disorder. Using the equivalence to single-step growth models, the exact line energy of maximum energy paths is computed for exponential and geometric disorder distributions. The resulting expression depends on the distributions only through their mean and variance. A universal square-root singularity appears in the ground-state energy as the line orientation approaches the edges of the wedge-shaped lattice. We also discuss the relationship between point-to-point and point-to-line optimization problems, as well as their connection to the classic Wulff construction for crystal shapes. In addition, we derive some general, distribution-independent properties of the optimal path angle in the point-to-line configuration. This angle is argued to be directly measurable in fracture experiments of anisotropic materials such as paper.