Diffusion of random walk in a random environment.

We consider a continuous time random walk on the d-dimensional integer lattice in an environment which is random in space and time. The random walk may also be referred to as a directed polymer. In spatial integer dimension d greater than two and for sufficiently small random environment, we show that the behavior is diffusive with probability one. The diffusion constant is renormalised and is determined by the averaged process. The averaged process is given by averaging over the random environment initially. An investigation into whether the behavior is diffusive on average leads to the result that all the higher moments of the partition function are infinite, regardless of the dimension or the size of the random environment.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/104606/1/9542818.pdfDescription of 9542818.pdf : Restricted to UM users only