Directed and diode percolation

Abstract

We study the novel percolation phenomena that occur in random-lattice networks consisting of resistor-like and diode-like bonds. Resistor bonds connect or "transmit information" in either direction along their length, while diodes connect in one direction only. We first treat the special case of directed bond percolation, in which the diodes are aligned along a preferred axis. Mean-field theory shows that clusters become extremely anisotropic near the percolation transition and that their shapes are characterized by two correlation lengths, one parallel and one transverse to the preferred axis. These lengths diverge with exponents ${\nu }_{\parallel }=1\mathrm{}$ and ${\nu }_{\perp }=\frac{1}{2}$, respectively, from which we can show that the upper critical dimension for this system must be five. We also treat a more general random network on the square lattice containing resistors and diodes of arbitrary orientation. Duality arguments are applied to obtain exact results for the location of phase transitions in this system. We then use a position-space renormalization-group approach to map out the phase diagram and calculate critical exponents. This system has an isotropic percolating phase, and phases which percolate in only one direction. Novel types of transitions occur between these phases, in which the diode orientation plays a fundamental role. These percolating phases meet with the nonpercolating phase along a line of multicritical points, where concentration and orientational fluctuations are simultaneously critical.