Guo, Xiaoqin2013-01-102013-01-102012-09http://purl.umn.edu/142482University of Minnesota Ph.D. dissertation. September 2012. Major: Mathematics. Advisor: Ofer Zeitouni. 1 computer file (PDF); iv, 104 pages, appendix A.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.en-USEinstein relationInvariance principleLimiting velocityRandom walk in random environmentThesis or Dissertation