© 2015 American Physical Society. We report electrical conductivity σ measurements on a range of two-dimensional electron gases (2DEGs) of varying linear extent. Intriguingly, at low temperatures (T) and low carrier density (ns) we find the behavior to be consistent with σ∼Lα, where L is the length of the 2DEG along the direction of transport. Importantly, such scale-dependent behavior is precisely in accordance with the scaling hypothesis of localization [Abrahams, Phys. Rev. Lett. 42, 673 (1979)PRLTAO0031-900710.1103/PhysRevLett.42.673] which dictates that in systems where the electronic wave function ξ is localized, σ is not a material-specific parameter but depends on the system dimensions. From our data we are able to construct the "β function" ≡(h/e2)dlnσ/dlnL and show this to be strongly consistent with theoretically predicted limiting values. These results suggest, remarkably, that the electrons in the studied 2DEGs preserve phase coherence over lengths ∼10μm. This suggests the utility of the 2DEGs studied towards applications in quantum information as well as towards fundamental investigations into many-body localized phases.