Law of the Iterated Logarithm for the random walk on the infinite   percolation cluster
  We show that random walks on the infinite supercritical percolation clusters in Z^d satisfy the usual Law of the Iterated Logarithm. The proof combines Barlow's Gaussian heat kernel estimates and the ergodicity of the random walk on the environment viewed from the random walker as derived by Berger and Biskup. 
H. Duminil-Copin