Abstract
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.
Highlights
Introduction and main resultsThis paper is a continuation of our earlier work [16]
We studied upper deviation estimates for the random walk in random scenery in the case where the random scenery is non-negative and has a Pareto distribution
We give the precise estimates in the first regime (Theorem 1.1), that is needed for the asymptotic of the moderate deviation and the Green function for the random conductance model explained below, and a partial result for the second problem of lower deviation (Proposition 1.2)
Summary
This paper is a continuation of our earlier work [16]. In that paper, we studied upper deviation estimates for the random walk in random scenery in the case where the random scenery is non-negative and has a Pareto distribution. We give the precise estimates in the first regime (Theorem 1.1), that is needed for the asymptotic of the moderate deviation and the Green function for the random conductance model explained below, and a partial result for the second problem of lower deviation (Proposition 1.2). One of our main motivation is to show that in contrast with the standard situation, where the random conductance are independent and identically distributed (cf [4]), the upper tail of distribution of the conductance in a layered structure yields an anomalous behavior of the heat kernel and the Green function. To the random walk in random scenery, we found a power law decay regime in [16], and in this paper the sharp asymptotics is determined (Theorem 1.6). In contrast with the spectral dimension, the decay of the Green function exhibits a non-standard behavior even for finite mean conductance when d ≥ 5
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