Motivated by Radio Frequency (RF) heating studies, the response of the plasma of tokamaks to the presence of a locally polarized limiter is studied. In a first part, we use the TOKAM3X 3D global edge turbulence code to analyse the impact of such biasing in a realistic geometry. Key features of experimental observations are qualitatively recovered, especially the extension of a potential and density perturbation on long, but finite, distances along connected field lines. The perturbation is also found to extend in the transverse direction. Both observations demonstrate the influence of perpendicular current loops on the plasma confirming the need for an accurate description in reduced models. In a second part, we use the TOKAM2D slab turbulence code to determine the validity of using a transverse Ohm's law for this purpose. Results indicate that a local Ohm's law with a constant and uniform perpendicular resistivity appears at least as an oversimplified description of perpendicular charge transport in a turbulent Scrape-Off Layer.