ON 3 CONJECTURES BY SHULER,K.E.

Some fifteen years ago, Shuler formulated three conjectures relating t o the large-time asymptotic properties of a nearest-neighbor random wa lk on Z2 that is allowed to make horizontal steps everywhere but verti cal steps only on a random fraction of the columns. We give a proof of his conjectures for the situation where the column distribution is st ationary and satisfies a certain mixing condition. We also prove a str ong form of scaling to anisotropic Brownian motion as well as a local limit theorem. The main ingredient of the proofs is a large-deviation estimate for the number of visits to a random set made by a simple ran dom walk on Z. We briefly discuss extensions to higher dimension and t o other types of random walk.