Percolation and conduction in a random resistor-diode network

We study percolation and conduction in a randomly diluted network in which the occupied bonds may be either resistor-like, transmitting connectivity in both directions along a bond, or diode-like, transmitting in only one direction. This system exhibits novel phase transitions signalling the onset of infinite clusters with either isotropic or unidirectional connectivity. Position-space renormalisation group methods are applied to map out the phase diagram and calculate exponents associated with these phase transitions. The percolation problem has been extensively investigated, partly because it is an extremely simple system exhibiting the intriguing complexities of continuous phase transitions, apd also because of its many physical realisations (Stauffer 1979, Essam 1980). Recent work has focused on developing generalisations of percolation which are of fundamental theoretical interest, as well as models which more realistically describe the properties of particular physical systems. Many such generalisations are contained implicitly in the pioneering work of Broadbent and Hammersley (1957). They proposed that neighbouring lattice sites may be joined by two randomly occupied directed bonds, one ‘transmitting’ connectivity or information in one direction, and the other transmitting in the reverse direction. In this sense, the directed bonds act as diodes, in contrast to pure bond percolation in which the bonds act as resistors, transmitting in both directions. One special case of this Broadbent-Hammersley model is directed bond percolation, in which, on the square lattice, randomly occupied directed bonds may transmit only upward or to the right. Above the percolation threshold, an infinite cluster forms which may be traversed from the lower-left to the upper-right, but not vice versa. This model has different critical behaviour from pure bond percolation (Blease 1977a, b, c, KertCsz and Vicsek 1980, Dhar and Barma 1981), exhibiting an anisotropic structure for both the infinite cluster and the decay of correlations (Obukhov 1980, Kinzel and Yeomans 1981). Further interest in this model stems from the fact that directed percolation can be mapped into a Reggeon field theory (Cardy and Sugar 1980), and the latter model can then be related to Markov processes with absorption, branching and recombination (Grassberger and Sundermeyer 1978, Grassberger and de la Torre 1979) which are of relevance for describing many chemical and biological processes (Schlogl 1972). Reynolds (1981) treated a more general problem in which the diodes could ‘break down’, and conduct in both directions. By varying such a breakdown t Supported in part by grants from the ARO and AFOSR. 0305-4470/81/090349 + 06$01.50 @ 1981 The Institute of Physics L349 L350 Letter to the Editor probability, Reynolds studied the crossover between directed and isotropic percolation, and argued that the two models are in different universality classes. In this Letter, we consider a more general percolation process on the square lattice, mediated by both resistors and randomly oriented diodes. We define positive diodes to be transmitting either upward or to the right and vice versa for negative diodes. Resistors transmit in both directions, and vacancies are non-transmitting (figure 1).