One-dimensional lattice random walks in a Gaussian random potential

Abstract We study random walks evolving in continuous time on a one-dimensional lattice where each site x hosts a quenched random potential U x . The potentials on different sites are independent, identically distributed Gaussian random variables. We analyze three distinct models that specify how the transition rates depend on U x : the random-force-like model, random walks with randomized stepping times, and the Gaussian trap model. Our analysis focuses on five key disorder-dependent quantities defined for a finite chain with N sites: the probability current, its reciprocal (the resistance), the splitting probability E − , the mean first-passage time T N , and the diffusion coefficient D N in a periodic chain. By determining the moments of these random variables, we demonstrate that the probability current and resistance are not self-averaging, which leads to pronounced differences between their average and typical behaviors. In contrast, E − , T N and D N become self-averaging when N → ∞ , though they exhibit strong sample-to-sample fluctuations for finite N .