Nonlinear random resistor diode networks and fractal dimensions of directed percolation clusters

Abstract

We study nonlinear random resistor diode networks at the transition from the non percolating to the directed percolating phase. The resistor-like bonds and the diode-like bonds under forward bias voltage obey a generalized Ohm's law, $V \sim I^r$. Based on general grounds as symmetries and relevance we develop a field theoretic model. We focus on the average two-port resistance, which is governed at the transition by the resistance exponent $ϕ_r$. By employing renormalization group methods we calculate $ϕ_r$ for arbitrary $r$ to one-loop order. Then we address the fractal dimensions characterizing directed percolation clusters. Via considering distinct values of the nonlinearity $r$, we determine the dimension of the red bonds, the chemical path and the backbone to two-loop order.