A connection between linear and nonlinear resistor networks

We explore the connection between the higher moments of the current (or voltage) distribution in a random linear resistor network, and the resistance of a nonlinear random resistor network. We find that the two problems are very similar, and that an infinite set of exponents are required to fully characterise each problem. These exponent sets are shown to be identical on a particular hierarchical lattice, a simple model which accurately describes the geometrical properties of the backbone of the infinite cluster at the percolation threshold and also the voltage distribution on this structure. The critical behaviour of random resistor networks has been extensively studied in the past few years (see e.g., Zabolitsky 1984, Herrmann er a1 1984, Hong et a1 1984, Lobb and Frank 1984, and references therein). However, it has been only very recently that attention has turned to the distribution of voltage drops across each conductor in a resistor network (de Arcangelis et al 1985). The importance of this distribution is that it provides detailed microscopic information about the structure of the network, in addition to providing fundamental information such as the network conductivity. Our goal, in this letter, is to point out a connection between the higher moments of the voltage distribution on linear resistor networks and the resistance of nonlinear networks. To define the voltage distribution on a linear network, consider a hypercubic cell of linear dimension L which contains a random resistor network at the percolation threshold. If a unit voltage drop is applied across opposite faces of the network, a total current I,,, will flow. Since the voltage drop V between opposite faces is unity, I,,, simply equals G, where G is the conductance of the system. Furthermore, notice that I,,, coincides with the current flowing through the links, or cutting bonds of the backbone. These are defined as the bonds, which, if cut, cause the opposite faces of the network to become disconnected. Each bond of the network can be characterised by the fraction of the total current flowing through it, a = Z/ZtOt. For example, the cutting bonds are characterised by a = 1, while bonds which belong to very large blobs are mostly characterised by very small values of a. An interesting feature of this bond characterisation is that an infinite set of independent lengths, &, and corresponding exponents, &., can be defined at the percolation threshold via s k = c akN(a)-Lik (1) t Supported in part by grants from the ARO, NSF and ONR. 0305-4470/85/130805 + 04$02.25