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 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‖ , where t 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‖/k, where k is the strength of the disorder. These results are supported by numerical simulations. Copyright c © EPLA, 2007 The directed polymer model [1] is a well-studied [2] model in the field of disordered systems. The model is concerned with directed optimal paths in random media, which are characterized by two growth rate exponents, ω and ν. The exponent ω determines the energy variability of a path of length t by the relation ∆E ∼ t, and the exponent ν determines the mean transversal distance of the optimal paths from the origin, through the relation D∼ t . These two growth rate exponents are connected by the Huse-Henley scaling relation: ω= 2ν− 1 [3]. There are two cases in which the space exponent ν has the value of the directed percolation model [4], rather than its value in the regular case. In the first case [5,6], the bonds of the lattice are assigned with values taken from a bimodal (0,1) distribution, and the probability to have a zero-valued bond is pc, the critical probability of directed percolation. In the second case, the bonds are assigned with values taken from a strong disordered distribution, and thus the energy of each path, which is defined as the sum of its bonds’ values, is mainly determined by the value of the maximal bond along that path. In the strong disorder limit, the optimal paths are identical to the optimal Min-Max paths, which are characterized by the directed percolation exponent ν [7,8]. Directed Min-Max optimal paths are the subject of the present article, which studies two cases in which there is a small probability for a change in the position of the optimal path. In the first case, small perturbations are applied to the weights of the bonds of the lattice, and the new Min-Max optimal path might be different from the original one. This case was numerically studied in [9] for directed polymer (regular) optimal paths (which are determined by the minimal sum of bond values), and a general theoretical discussion was presented in [10]. Explanations to the numerical results presented in [9] were given in [10–13]. The second case is the one of strong disorder: While in the strong disorder limit the regular optimal path is identical to the Min-Max optimal path, as the strength of the disorder decreases, a strong disorder weak disorder transition occurs. This transition was studied for the ordinary (non-directed) lattice in [14,15], and for the directed case in [16]. The present study shows that in the case of small perturbations, the probability to depart from the original Min-Max optimal path is