Central limit theorem for random walks in fluctuating environments

In the current paper, the authors extend earlier results for a one-particle random walk. First they prove that X_t obeys a local central limit theorem even if the interactions between the walk and the field occur over a finite region containing X_t. Interactions between the walk and the field are still sufficiently small. Secondly they prove that in a special case where |S|=2 and interactions are again strictly local, a local central limit theorem holds with the smallness of the interactions replaced by a more general nondegeneracy condition. The proofs of the results rely on cluster expansion techniques.