| We consider some questions in the area of random walks in random environment (RWRE). We first consider an i.i.d. random environment with a strong form on transience on the two dimensional integer lattice. We prove a functional central limit theorem (CLT) of the quenched expected position of the random walk indexed by its level crossing times.;Our next problem considers a system of particles on the d dimensional integer lattice evolving in a dynamical random environment. We prove that the invariant distributions for this system consists of mixtures of independent Poisson measures. For the case d = 1, 2, we further show that these are the only invariant distributions and we also show the convergence to these starting from any initial ergodic distribution. For the case d = 1, we also study the fluctuations of the current as observed by an observer moving at the average speed of the walks. We get that the fluctuations of the current centered about the quenched mean are of order n¼.;We also obtain asymptotics of the Green's functions of perturbed symmetric random walks by comparing them to the Green's functions of the corresponding unperturbed walks. |
