Diffusion in sparse networks: linear to semilinear crossover. Academic Article uri icon

abstract

  • We consider random networks whose dynamics is described by a rate equation, with transition rates ${w}_{nm}$ that form a symmetric matrix. The long time evolution of the system is characterized by a diffusion coefficient $D$. In one dimension it is well known that $D$ can display an abrupt percolation-like transition from diffusion ($D>0$) to subdiffusion ($D=0$). A question arises whether such a transition happens in higher dimensions. Numerically $D$ 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 $D$ is finite, suggest an ``effective-range-hopping'' procedure to evaluate it, and contrast the results with the linear estimate. The same approach is useful in the analysis of networks that are described by quasi-one-dimensional sparse banded matrices.

publication date

  • January 1, 2012