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Multifractal properties of the random resistor network

Barthelemy¹, Buldyrev, Havlin

¹Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

|November 23, 2000

Summary

This study reveals that in 2D random resistor networks at percolation, small currents follow a P(i) ~ 1/i distribution. This implies moments of current do not exist, with low currents sharing the backbone

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Area of Science:

Condensed matter physics
Statistical physics
Network science

Background:

Percolation theory describes the behavior of connected systems near a critical threshold.
Random resistor networks are fundamental models in statistical physics.

Purpose of the Study:

Investigate the multifractal spectrum of electrical current in 2D random resistor networks at the percolation threshold.
Analyze current distribution under different voltage application methods (parallel bars vs. points).

Main Methods:

Numerical simulations of two-dimensional random resistor networks.
Analysis of current probability distributions and their moments.
Characterization of fractal dimensions of network components.

Main Results:

The probability distribution of small currents follows P(i) ~ 1/i in the infinite system limit.
Moments of the current of order q <= 0 do not exist.
Low currents exhibit the fractal dimension of the network backbone.

Conclusions:

The current distribution in these networks is highly heterogeneous.
The network backbone can be characterized by fractal blobs and high-current carrying bonds.
Understanding current flow is crucial for disordered systems at criticality.