Optimal paths in strong and weak disorder: a unified approach

We present a unified scaling theory for the optimal path connecting opposite edges of a disordered lattice of size L. Each bond of the lattice is assigned a cost exp(ar), where r is a uniformly distributed random variable and a is disorder strength. The optimal path minimizes the sum of the costs of the bonds along the path. We argue that for L>>a(nu) , where nu is the correlation exponent of percolation, the path becomes equivalent to a directed polymer on an effective lattice consisting of blobs of size xi=a(nu). It is self-affined and characterized by the roughness exponent of directed polymers chi. For L<