Study of Kalikow Problem

Random walk’in a random environment, a process where a randomly chosen transition probability is assigned to each point, has been exhausivly studied in the one dimensional nearest neighbor case but is much more difficult in any more general case. I-n particular, in the two dimensional case little thing is known exept Theorem 1 0f （1）. It is far from optimal and Kailikow proposed Problem 4 of （1）, i.e., one of the below two environment is chosen to each point of Z² in an i.i.d. manner woth the same probability.Is this R.W.I.R.E. recurent? The conclusion of Problem 4 of （1） seems obvious, but none of the previous, techniques seems to handl it exhausively. Conbining the technique of Number Theory with the general methods of Probability theory, we get that R.W.I.R.E. satisfying （*） is transient.