Moderate deviations for the range of planar random walks
  Given a symmetric random walk in $Z^2$ with finite second moments, let $R_n$ be the range of the random walk up to time $n$. We study moderate deviations for $R_n -E R_n$ and $E R_n -R_n$. We also derive the corresponding laws of the iterated logarithm. 
Richard F. Bass, Xia Chen, and Jay Rosen