Green's Functions For Random Resistor Networks

TL;DR

This paper develops a Green's function approach to analyze random resistor networks, providing a systematic perturbation expansion, explicit solutions for small disorder cases, and a new order parameter to characterize disorder regimes.

Contribution

It introduces a systematic disorder perturbation expansion, a recursive formalism for exact Green's functions with disordered bonds, and a novel order parameter for disorder characterization.

Findings

Validated perturbation expansion with numerical simulations

Derived explicit Green's functions for up to four disordered bonds

Proposed an order parameter to distinguish disorder regimes

Abstract

We analyze random resistor networks through a study of lattice Green's functions in arbitrary dimensions. We develop a systematic disorder perturbation expansion to describe the weak disorder regime of such a system. We use this formulation to compute ensemble averaged nodal voltages and bond currents in a hierarchical fashion. We verify the validity of this expansion with direct numerical simulations of a square lattice with resistances at each bond exponentially distributed. Additionally, we construct a formalism to recursively obtain the exact Green's functions for finitely many disordered bonds. We provide explicit expressions for lattices with up to four disordered bonds, which can be used to predict nodal voltage distributions for arbitrarily large disorder strengths. Finally, we introduce a novel order parameter that measures the overlap between the bond current and the optimal path (the path of least resistance), for a given resistance configuration, which helps to characterize the weak and strong disorder regimes of the system.