Multiscaling approach in random resistor and random superconducting networks
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Abstract
We report on a variety of novel features for the distribution of voltage drops across the bonds of a random resistor network. To describe this distribution analytically, we introduce a simple geometrical model, with a hierarchical structure of links and blobs, which appears to capture the basic features of random networks near the percolation threshold. On this model, we find that the voltage distribution is a log binomial, and that an infinite hierarchy of exponents is required to characterize the moments of this distribution. On general grounds, we argue that this exponent hierarchy emerges naturally from an underlying distribution which, at the percolation threshold, can be written in the form, ${L}^{\ensuremath{\varphi}(\mathrm{l}\mathrm{n}\mathit{V}/\mathrm{l}\mathrm{n}{V}_{\mathrm{max})}}$, where L is the linear size of the system, V is the voltage drop, and ${V}_{\mathrm{max}}$ is the maximum value of this voltage drop. The nonconstancy of \ensuremath{\varphi}(y) as a function of y is an unconventional feature in the context of a scaling approach, and a variety of novel properties result. These are tested by numerical simulations of the voltage distribution for square-lattice networks at the percolation threshold. In particular, the moments of this distribution are found to scale independently, with the exponents of the positive moments in excellent agreement with those of the hierarchical model. We also discuss some intriguing properties associated with the voltage distribution above the percolation threshold, most notably, that the higher moments of the distribution are nonmonotonic functions of the bond concentration. Finally, we exploit duality arguments to investigate the voltage distribution of a random superconducting network.