Stability of directed Min-Max optimal paths

The stability of directed Min-Max optimal paths in cases of change in the random media is studied. Using analytical arguments it is shown that when small perturbations (epsilon) are applied to the weights of the bonds of the lattice, the probability that the new Min-Max optimal path is different from the original Min-Max optimal path is proportional to t(1/nu)parallel to epsilon, where t is the size of the lattice, and nu(parallel to) is the longitudinal correlation exponent of the directed percolation model. It is also shown that in a lattice whose bonds are assigned with weights which are near the strong disorder limit, the probability that the directed polymer optimal path is different from the optimal Min-Max path is proportional to t(2/nu)parallel to/k(2), where k is the strength of the disorder. These results are supported by numerical simulations. Copyright (C) EPLA, 2007.