A General Framework for Consistent Estimation of Charge Transport Properties via Random Walks in Random Environments

A general framework is proposed for the study of the charge transport properties of materials via Random Walks in Random Environments (RWRE). The material of interest is modelled by a random environment and the charge carrier is modelled by a random walker. The framework combines a model for the fast generation of random environments that realistically mimic materials morphology with an algorithm for efficient estimation of key properties of the resulting random walk. The model of the environment makes use of tools from spatial statistics and the theory of random geometric graphs. More precisely, the disordered medium is represented by a random spatial graph with directed edge weights, where the edge weights represent the transition rates of a Markov Jump Process (MJP) modelling the motion of the random walker. This MJP is a multiscale stochastic process. In the long term, it explores all vertices of the random graph model. In the short term, however, it becomes trapped in small subsets of the state space and makes many transitions in these small regions. This behaviour makes efficient estimation of velocity by Monte Carlo simulations a challenging task. Therefore, we use Aggregate Monte Carlo (AMC), introduced in Brereton et al. (2014), for estimating the velocity of a random walker as it passes through a realisation of the random environment. In this paper, we prove the strong consistency of the AMC velocity estimator and use this result to conduct a detailed case study, in which we describe the motion of holes in an amorphous mesophase of an organic semiconductor, dicyanovinyl-substituted oligothiophene (DCV4T). In particular, we analyse the effect of system size (i.e. number of molecules) on the velocity of single charge carriers.