The Resistance of Feynman Diagrams and the Percolation Backbone Dimension

TL;DR

This paper introduces a novel interpretation of Feynman diagrams as resistor networks to simplify the field theory of percolation transport, enabling precise calculation of the fractal dimension of the percolation backbone.

Contribution

It provides a new perspective on Feynman diagrams for percolation, leading to an advanced three-loop order calculation of the backbone's fractal dimension using renormalization group techniques.

Findings

Calculated the backbone fractal dimension $D_B$ to three-loop order.

Derived an epsilon expansion formula for $D_B$ involving zeta function.

Simplified the field theory of transport on percolation clusters.

Abstract

We present a new view of Feynman diagrams for the field theory of transport on percolation clusters. The diagrams for random resistor networks are interpreted as being resistor networks themselves. This simplifies the field theory considerably as we demonstrate by calculating the fractal dimension $D_B$ of the percolation backbone to three loop order. Using renormalization group methods we obtain $D_B = 2 + \epsilon /21 - 172\epsilon^2 /9261 + 2 \epsilon^3 (- 74639 + 22680 \zeta (3))/4084101$, where $\epsilon = 6-d$ with $d$ being the spatial dimension and $\zeta (3) = 1.202057..$.