Percolation properties of a three-dimensional random resistor-diode network

We study percolation in a random resistor-diode network on the simple cubic lattice. Here the occupied bonds joining nearest-neighbour sites may be either resistor-like, transmitting information or connectivity in either direction along a bond, or diode-like, transmitting in only one direction. We consider a model in which there exist both positive diodes which transmit in the fx, + y or + z directions, and negative diodes which transmit oppositely. We apply position-space renormalisation group methods to map out the phase diagram, and calculate the exponents associated with the phase transitions of this system. We find three novel types of transitions due to the presence of the diodes. With only one species of diodes and vacancies (unoccupied bonds) present, a directed threshold occurs at a critical concentration of diodes. Here an infinite cluster forms which transmits only in the direction of the average diode polarisation. A reverse threshold occurs when resistors and only one species of diodes exist. At this point the resistors mediate information flow in the infinite cluster opposite to the diode polarisation. Finally, with all bond elements present, a mixed threshold occurs. At this point, an isotropic infinite cluster exists, but of a qualitatively different character from that occurring at the usual bond percolation threshold. The mixed and isotropic transitions are higher-order critical points where diode, resistor and vacancy phases become simultaneously critical. The percolation problem has been extensively investigated, partly because it is an extremely simple system exhibiting the intriguing complexities of second-order phase transitions, and also becuse of the many realisations of percolation phenomena in nature. (See e.g. Stauffer (1979), Essam (1980) for recent reviews.) Recently, attention has focused on developing more general percolation models which are a challenge on a fundamental theoretical level, as well as finding applications for percolation in many diverse fields. Many such generalisations are contained implicitly in the early work of Broadbent and Hammersley (1957). They proposed a percolation process in which 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, thus breaking the isotropic symmetry of the usual bond percolation problem. One special case of this Broadbent-Hammersley percolation process is directed bond percolation. For example, on the square lattice, the occupied bonds may transmit only upward or to the right. Above the percolation threshold, the infinite cluster can be traversed from the lower left to the upper right, but not in the reverse direction. This t Supported in part by grants from the ARO and AFOSR. 0305-4470/81/080285 + 06$01.50 @ 1981 The Institute of Physics L285 L286 Letter to the Editor model has several interesting features not found in isotropic bond percolation. These include different critical behaviour (Blease 1977a, b, c, Kertksz and Vicsek 1980), and an anisotropic structure for the infinite cluster (Obukhov 1980). Because of the latter result, the decay of correlations is characterised by two length scales, one parallel and one perpendicular to the directed axis iDhar and Barma 1981, Kinzel and Yeomans 1981). Directed percolation has been mapped into a Reggeon field theory (Cardy and Sugar 1980), and the latter model can be related to Markov processes with absorption, branching and recombination (Grassberger and Sundermeyer 1978, Grassberger and de la Torre 19791, which are of relevance for describing many chemical and biological processes (Schlogl 1972). Very recently, there has been an interest in generalising directed bond percolation to a situation where the orientation of the diodes is random. In this connection, Reynolds (1981) considered a model in which a lattice was randomly occupied by either positive diodes (transmitting upward or to the right), or negative diodes (transmitting in the opposite direction). In addition, each of these diodes could 'break down' with a random probability and transmit in both directions. Redner (1981) treated a similar problem of a random resistor-diode network. In this model, each bond may be independently occupied by positive or negative diodes, resistors and vacancies (empty bonds). In both of these systems, the possibility of continuously varying the diode polarisation leads to a rich phase diagram and novel types of phase transitions. These include thresholds for the formation of an infinite cluster which transmits information unidirectionally, either parallel or antiparallel to the diode polarisation. In addition, there exist isotropic thresholds which exhibit higher-order critical behaviour where resistor, vacancy and two diode phases become simultaneously critical. Because of the richness of this model, we have extended the study of the physical properties of the random resistor-diode network to the simple cubic lattice. In this Letter, we consider a model system containing positive diodes which transmit in the +x, + y or +t directions, and negative diodes which transmit in the opposite direction. Resistors transmit in either direction, and vacancies are non-transmitting (see figure l ( a ) ) . These elements occur with random probabilities d,, d-, r and o respectively. In this model, the diode polarisation points along the (1, 1, 1) axis, and can take any value between -1 and +l. When the polarisation per occupied bond equals i l , we recover directed percolation. On the other hand, when the polarisation equals zero, we have a