Estimating hitting probabilities of an interacting particle system on a graph

We consider the problem of estimating hitting probabilities related to a class of interacting particle systems. These systems, in which two types of particles — ‘electrons’ and ‘holes’ — move on a graph, are simplified versions of models describing charge transport in disordered materials. The probability of interest is the probability that an electron reaches a certain region of the graph before colliding with a hole. We provide a detailed description of our model and explain how it can be simulated. Next, we give a brief introduction to importance sampling, which we use to improve the efficiency of our estimators. To our knowledge, importance sampling has not yet been used to estimate probabilities related to interacting particle systems. We describe how importance sampling can be used to improve the efficiency of estimators of hitting time probabilities involving discrete time Markov chains. We then use importance sampling to estimate the probability that we are interested in. In doing so, we observe that there are a number of complexities that arise when working with interacting particle systems. We describe some simple heuristics for implementing importance sampling. These heuristics make minimal changes to the probability measure under which the original system evolves. We consider a specific example of our problem and investigate the effectiveness of the importance sampling approach. We show that our estimators outperform standard Monte Carlo estimators. Finally, we describe possible future work, which includes a more sophisticated importance sampling approach that uses ‘locally optimal’ changes of measure.