Effective dynamics of a random walker on a heterogeneous ring: Exact results

In this paper, by considering a biased random walker hopping on a one-dimensional lattice with a ring geometry, we investigate the fluctuations of the speed of the random walker. We assume that the lattice is heterogeneous i.e. the hopping rate of the random walker between the first and the last lattice sites is different from the hopping rate of the random walker between the other links of the lattice. Assuming that the average speed of the random walker in the steady-state is v∗, we have been able to find the unconditional effective dynamics of the random walker where the absolute value of the average speed of the random walker is −v∗. Using a perturbative method in the large system-size limit, we have also been able to show that the effective hopping rates of the random walker near the defective link are highly site-dependent.