Conductivity of random resistor-diode networks

We investigate the behavior of the conductivity in random lattice networks of resistors and Ohmic diodes—elements which behave like an ideal resistor in the forward direction. We study the case where the orientation of the diodes is fixed, corresponding to the geometrical model of directed percolation. New critical behavior is predicted because the structure of the underlying conducting network is characterized by two independent orthogonal correlation lengths. We use the node picture of a percolating network to derive the anisotropic scaling relation between directed percolation and conductivity exponents. In the mean-field limit, we find a directed conductivity exponent $t_{+}=2\mathrm{}$, in contrast to an isotropic conductivity exponent of $t=3\mathrm{}$. In two dimensions, we employ the renormalization group to study the critical behavior of the directed conductivity. We predict that the conductivity should approach zero with an infinite slope ($t_{+}<1\mathrm{}$) as the transition is approached from above. This is consistent with the intuitive expectations developed from the node picture. We also briefly discuss conduction in networks with superconducting diodes (which have an infinite conductivitivity in the forward direction), and possibility of a directed conductor-superconductor transition. Additionally, we describe a more general network with Ohmic and superconducting diodes within the renormalization-group framework by introducing a larger parameter space which includes "leaky" diodes.