Exact exponent relations for random resistor-diode networks

We investigate the percolation properties of a random network of non-directed bonds (resistors) and urbitrurily oriented directed bonds (diodes). For the square lattice, there is a multicritical line which connects the isotropic percolation threshold with a network fully occupied by randomly oriented diodes (‘random Manhattan’). Along this line, symmetry and invariance properties are used to demonstrate that the correlation length exponents are constant. Furthermore, a lattice independent relation is proved which shows that at the isotropic percolation threshold, the correlation length diverges with the same exponent when the transition is approached by varying either the resistor concentration or the concentration of randomly oriented diodes. Recently, there has been considerable attention devoted to directed percolation, a geometrical model in which directed bonds of a fixed orientation connect lattice sites (for a current review see e.g. Kinzel(l982)). Much of the interest in this model stems from its novel anisotropic critical behaviour. Applications of directed percolation include Reggeon field theory (Cardy and Sugar 1980) and branching Markov processes (Schlogl 1972) which arise in chemical reaction models. In this letter, we consider a more general percolation model containing directed bonds (diodes) of arbitrary orientation as well as non-directed bonds (resistors). Our motivation for this study arises in part because this model has much richer geometrical behaviour than either directed or pure isotropic percolation since connected paths mediated by an arbitrary combination of resistors and diodes can form (Redner 1981, 1982). In addition, this resistor-diode percolation model may be relevant for describing various aspects of information theory and communication network problems (see e.g. Ford and Fulkerson 1962, Frank and Frisch 1971). The primary results of this work are to derive general relations for the correlation length exponents of the network. In particular, for the square lattice we shall demonstrate that correlation length exponents are constant along a multicritical line in the phase diagram. In addition, at the isotropic percolation threshold, the correlation length diverges with the same exponent if the transition is approached by varying either the resistor concentration or the concentration of randomly oriented diodes. This second result is valid for all lattices. Both exponent relations are obtained within the framework of the position-space renormalisation group (PSRG) with an arbitrary rescaling factor. Since the PSRG is expected to become exact in the infinite cell-size limit (Reynolds et a1 1980), our results for the exponents should become exact as well. t Supported in part by grants by the ARO, NSF and ONR. 0305-4470/82/120685 + 06$02.00 @ 1982 The Institute of Physics L685 L686 Letter to the Editor On the square lattice, the random resistor-diode network is defined by joining nearest-neighbour sites by a positive diode (conducting either upward or to the right), a negative diode (conducting in the opposite direction), a resistor, or the sites may be disconnected with respective probabilities p + , p , p and q = 1 p + p p (figure l(a)). From a previous small-cell PSRG calculation, an extremely symmetric phase diagram was obtained (figure l(6)). Our interest is primarily on the multicritical line where four phases in the system are simultaneously critical (figure l(c)) . This line is defined by both reflection symmetry ( p + = p ) and dual symmetry ( p = 4). One end