Diffusion in sparse networks: Linear to semilinear crossover

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.