Diffusion in sparse networks: linear to semi-linear crossover
arXiv:1206.2495 · doi:10.1103/PhysRevE.86.051120
Abstract
We consider random networks whose dynamics is described by a rate equation, with transition rates that form a symmetric matrix. The long time evolution of the system is characterized by a diffusion coefficient . In one dimension it is well known that can display an abrupt percolation-like transition from diffusion () to sub-diffusion (D=0). A question arises whether such a transition happens in higher dimensions. Numerically can be evaluated using a resistor network calculation, or optionally it can be deduced from the spectral properties of the system. Contrary to a recent expectation that is based on a renormalization-group analysis, we deduce that is finite; suggest an "effective-range-hopping" procedure to evaluate it; and contrast the results with the linear estimate. The same approach is useful for the analysis of networks that are described by quasi-one-dimensional sparse banded matrices.
13 pages, 4 figures, proofed as published
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