Diluted networks of nonlinear resistors and fractal dimensions of percolation clusters.

Abstract

We study random networks of nonlinear resistors, which obey a generalized Ohm's law V approximately Ir. Our renormalized field theory, which thrives on an interpretation of the involved Feynman diagrams as being resistor networks themselves, is presented in detail. By considering distinct values of the nonlinearity r, we calculate several fractal dimensions characterizing percolation clusters. For the dimension associated with the red bonds we show that dred = 1/nu at least to order O(epsilon 4), with nu being the correlation length exponent, and epsilon = 6 - d, where d denotes the spatial dimension. This result agrees with a rigorous one by Coniglio. Our result for the chemical distance, dmin = 2 - epsilon/6 - [937/588 + 45/49(ln 2 - 9/10 ln 3)](epsilon/6)2 + O(epsilon 3) verifies a previous calculation by one of us. For the backbone dimension we find DB = 2 + epsilon/21 - 172 epsilon 2/9261 + 2[-74639 + 22680 zeta(3)]epsilon 3/4084101 + O(epsilon 4), where zeta(3) = 1.202057..., in agreement to second order in epsilon with a two-loop calculation by Harris and Lubensky.