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 ε 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/ν∥ _ε, wheret is the size of the lattice, and ν_|| 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/ν||/k², where k is the strength of the disorder. These results are supported by numerical simulations.