The 'emergent scaling' phenomenon and the dielectric properties of random resistor–capacitor networks

An efficient algorithm, based on the Frank–Lobb reduction scheme, for calculating the equivalent dielectric properties of very large random resistor–capacitor (R–C) networks has been developed. It has been used to investigate the network size and composition dependence of dielectric properties and their statistical variability. The dielectric properties of 256 samples of random networks containing: 512, 2048, 8192 and 32 768 components distributed randomly in the ratios 60% R–40% C, 50% R–50% C and 40% R–60% C have been computed. It has been found that these properties exhibit the anomalous power law dependences on frequency known as the 'universal dielectric response' (UDR). Attention is drawn to the contrast between frequency ranges across which percolation determines dielectric response, where considerable variability is found amongst the samples, and those across which power laws define response where very little variability is found between samples. It is concluded that the power law UDRs are emergent properties of large random R–C networks.