We present an experimental study of mesoscopic, two-dimensional electronic systems at high magnetic fields. Our samples, prepared from a low-mobility $\mathrm{I}\mathrm{n}\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}/\mathrm{I}\mathrm{n}\mathrm{A}\mathrm{l}\mathrm{A}\mathrm{s}$ wafer, exhibit reproducible, sample specific, resistance fluctuations. Focusing on the lowest Landau level, we find that, while the diagonal resistivity displays strong fluctuations, the Hall resistivity is free of fluctuations and remains quantized at its $\nu =1$ value, $h/e^2$. This is true also in the insulating phase that terminates the quantum Hall series. These results extend the validity of the semicircle law of conductivity in the quantum Hall effect to the mesoscopic regime.

• Received 17 May 2002

DOI:https://doi.org/10.1103/PhysRevLett.90.246802