Limit theorems for the Green function of the lattice Laplacian under large deviations of the random walk

We carry out a resolvent analysis of the lattice Laplacian (the generator of a simple random walk on the $d$-dimensional integer lattice) under large deviations of the random walk. This enables us to obtain asymptotic representations for the transition probability of the simple random walk and the corresponding Green function. We explicitly describe the asymptotic behaviour of the transition probability as the spatial and temporal variables jointly tend to infinity. The resulting Cramér-type expansion for the transition probability is ‘universal’ in this sense. In particular, it enables us to construct a scale for measuring the transition probability as a function of the time $t$ assuming that the spatial variable is of order $t^{\alpha}$ for various values of $\alpha\geqslant0$. We prove limit theorems on the asymptotic behaviour of the Green function of the transition probabilities under large deviations of the random walk.