Distribution functions for random walk processes on networks: An analytic method

Abstract

A novel analytic method for deriving and analyzing probability distribution functions of variables arising in random walk problems is presented. Applications of the method to quasi-one-dimensional systems show that the generating functions of interest possess simple poles, and no branch cuts outside the unit complex disk. This fact makes it possible to derive closed formulas for the full probability distribution functions and to analyze their properties. We find that transverse structures attached to a one-dimensional backbone can be responsible for the appearance of power laws in observables such as the distribution of first arrival times or the total current moving through a (model) photoexcited dirty semiconductor (our results compare well with experiment). We conclude that in some cases a geometrical effect, e.g., that of a transverse structure, may be indistinguishable from a dynamical effect (long waiting time); we also find universal shapes of distribution functions (humped structures) which are not characterized by power laws. The role of bias in determining properties of quasi-one-dimensional structures is examined. A master equation for generating functions is derived and applied to the computation of currents. Our method is also applied to a fractal structure, yielding nontrivial power laws. In all finite networks considered, all probability distributions decay exponentially for asymptotically long times.