  In this thesis, we study the diffusive and ballistic behaviors of random walk
in random environment (RWRE) in an integer lattice with dimension at least 2.
  Our contributions are in three directions: a conditional law of large numbers
and regeneration structures for RWRE in Gibbsian environments, quenched
invariance principles for balanced elliptic (but non uniformly elliptic)
environments, and a proof of the Einstein relation for balanced iid uniformly
elliptic environments.
