Weak disorder perturbation expansion for random resistance networks

Random resistance networks are a popular paradigm in statistical physics for modeling transport in disordered media such as semiconductors, spin glasses, and even porous rocks. We study a square lattice where the resistances at each bond are chosen from a statistical distribution. We derive the moments of the nodal voltages and bond currents exactly up to second order in a perturbation expansion in the disorder strength. Further, we construct an order parameter using the bond currents at arbitrary and infinite disorder, which provides a characterization of the behaviour of the system in the weak as well as strong disorder regimes. We predict a square law scaling with weak disorder, which is verified through numerical simulations. Scalings for the strong disorder regime are also obtained numerically, and the susceptibility of the order parameter is shown to diverge at a critical disorder. We compare our results with previous studies that use exponentially distributed resistances at each bond.