Nonlinear resistor fractal networks, topological distances, singly connected bonds and fluctuations

Abstract

The authors consider a fractal network of nonlinear resistors, with the voltage V behaving as a power of the current I, mod V mod =R mod I mod ^alpha . The resistance between two points at a distance L is R(L) varies as L^{zeta ( alpha )}. They prove that zeta (0) describes the scaling of the topological-chemical distance, while zeta ( infinity ) describes that of the number of singly connected 'red' bonds. For random resistors, they also consider the width of the resistance distribution, Delta R varies as L( zeta ₂( alpha )). Values for zeta and zeta ₂ are explicitly derived for two model fractals, and Delta R/R is found to grow with L for the Sierpinski gasket and alpha >1.612. The relevance of the results to percolation clusters is discussed.