  We report electrical conductivity $\sigma$ measurements on a range of
two-dimensional electron gases (2DEGs) of varying linear extent. Intriguingly,
at low temperatures ($T$) and low carrier density ($n_{\mathrm{s}}$) we find
the behavior to be consistent with $\sigma \sim L^{\alpha}$, 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~\textit{et al.}, Phys. Rev. Lett. \textbf{42}, 673
(1979)] which dictates that in systems where the electronic wave function $\xi$
is localized, $\sigma$ is not a material-specific parameter, but depends on the
system dimensions. From our data we are able to construct the
"$\beta$-function" $\equiv (h/e^2) d \ln \sigma / d \ln L$ 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 $\sim~10~\mu$m. This suggests the utility of the 2DEGs
studied towards applications in quantum information as well as towards
fundamental investigations into many-body localized phases.
