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 aν, where ν is the correlation exponent of percolation, the path becomes equivalent to a directed polymer on an effective lattice consisting of blobs of size ξ= aν. It is self-affined and characterized by the roughness exponent of directed polymers χ. For L aν, or on the length scales below the blob size ξ, the path behaves as an optimal path in the strong disorder limit. It has a self-similar fractal shape with fractal dimension dopt. We derived the scaling relations for the length of the path, its transversal displacement, the average cost and its fluctuation. We test our scaling theoretical predictions by numerical simulations on a square lattice.