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Directional decay of the Green’s function for a random nonnegative potential on \(\mathbb{Z}^d\). (English) Zbl 0938.60098
The author studies a simple random walk \((S_n)_{n\in\mathbb N_0}\) on \(\mathbb Z^d\) (in continuous or discrete time), moving in a random i.i.d. field of nonnegative random variables \(\omega(x), x\in\mathbb Z^d\), which is assumed to be independent of the walk. A transformed path measure is introduced via the density \(\exp(-\sum_{n=0}^{m-1} \omega(S_n))\) (suitably normalized). This random path measure can be interpreted in terms of first-passage site percolation and is a model for random electric networks or random walk with random killing. The author derives a shape theorem for the decay rate of the Green’s function for the transformed walk as the end-to-end distance goes to infinity and proves that the variance of its negative logarithm is linear in this distance. Furthermore, he proves a large-deviation principle for the endpoint of the path at a fixed time that goes to infinity, almost surely w.r.t. the environment. The author brings together methods used in earlier work by Sznitman on Brownian motion in a Poissonian potential with ideas from first-passage percolation. Most of the results can be seen as spatially discrete analogs of that work, but the methods of the present paper also cover the case that the starting point grows with the end-to-end distance (“uniform shape theorem”).
Reviewer: W.König (Berlin)

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses)
82B43 Percolation
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