Distribution functions in percolation problems

TL;DR

This paper uses renormalized field theory to analyze the asymptotic behavior of various probability distribution functions in percolation clusters, covering geometrical and transport properties.

Contribution

It provides a unified, diagrammatic approach to determine the asymptotic forms of distribution functions in percolation, valid at arbitrary loop order.

Findings

Derived asymptotic forms for pair-connection probability

Characterized distributions of backbone, red bonds, and self-avoiding walks

Analyzed distribution of total resistance in resistor networks

Abstract

Percolation clusters are random fractals whose geometrical and transport properties can be characterized with the help of probability distribution functions. Using renormalized field theory, we determine the asymptotic form of various of such distribution functions in the limits where certain scaling variables become small or large. Our study includes the pair-connection probability, the distributions of the fractal masses of the backbone, the red bonds and the shortest, the longest and the average self-avoiding walk between any two points on a cluster, as well as the distribution of the total resistance in the random resistor network. Our analysis draws solely on general, structural features of the underlying diagrammatic perturbation theory, and hence our main results are valid to arbitrary loop order.