Localization lengths for Schroedinger operators on Z^2 with decaying   random potentials
  We study a class of Schr\"odinger operators on $\Z^2$ with a random potential decaying as $|x|^{-\dex}$, $0<\dex\leq\frac12$, in the limit of small disorder strength $\lambda$. For the critical exponent $\dex=\frac12$, we prove that the localization length of eigenfunctions is bounded below by $2^{\lambda^{-\frac14+\eta}}$, while for $0<\dex<\frac12$, the lower bound is $\lambda^{-\frac{2-\eta}{1-2\dex}}$, for any $\eta>0$. These estimates "interpolate" between the lower bound $\lambda^{-2+\eta}$ due to recent work of Schlag-Shubin-Wolff for $\dex=0$, and pure a.c. spectrum for $\dex>\frac12$ demonstrated in recent work of Bourgain. 
Thomas Chen