Diluted Networks of Nonlinear Resistors and Fractal Dimensions of Percolation Clusters

H. K. Janssen; O. Stenull

doi:10.1103/PhysRevE.61.4821

paper

Diluted Networks of Nonlinear Resistors and Fractal Dimensions of Percolation Clusters

arXiv:cond-mat/9910427 · doi:10.1103/PhysRevE.61.4821

Abstract

We study random networks of nonlinear resistors, which obey a generalized Ohm's law, $V \sim I^{r}$ . 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 $d_{r e d} = 1/ ν$ at least to order ${\sl O} (ε^4)$, with $ν$ being the correlation length exponent, and $ε = 6 - d$ , where d denotes the spatial dimension. This result agrees with a rigorous one by Coniglio. Our result for the chemical distance, $d_{\scriptsize min} = 2 - ε/6 - [ 937/588 + 45/49 (\ln 2 -9/10 \ln 3)] (ε/6)^2 + {\sl O} (ε^3)$ verifies a previous calculation by one of us. For the backbone dimension we find $D_B = 2 + ε/21 - 172 ε^2 /9261 + 2 (- 74639 + 22680 ζ(3))ε^3 /4084101 + {\sl O} (ε^4)$, where $ζ (3) = 1.202057...$ , in agreement to second order in $ε$ with a two-loop calculation by Harris and Lubensky.

29 pages, 7 figures

Diluted Networks of Nonlinear Resistors and Fractal Dimensions of Percolation Clusters

Abstract

Cited by in corpus (22)