  We define a new class of random walk processes which maximize entropy. This
maximal entropy random walk is equivalent to generic random walk if it takes
place on a regular lattice, but it is not if the underlying lattice is
irregular. In particular, we consider a lattice with weak dilution. We show
that the stationary probability of finding a particle performing maximal
entropy random walk localizes in the largest nearly spherical region of the
lattice which is free of defects. This localization phenomenon, which is purely
classical in nature, is explained in terms of the Lifshitz states of a certain
random operator.
