This is a continuation of our earlier work [Stochastic Processes and their Applications, 129(1), pp.102–128, 2019] on the random walk in random scenery and in random layered conductance. We complete the picture of upper deviation of the random walk in random scenery, and also prove a bound on lower deviation probability. Based on these results, we determine asymptotics of the return probability, a certain moderate deviation probability, and the Green function of the random walk in random layered conductance.
"Quenched tail estimate for the random walk in random scenery and in random layered conductance II." Electron. J. Probab. 25 1 - 28, 2020. https://doi.org/10.1214/20-EJP478