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Published in:
2014 | OriginalPaper | Chapter
1. Random Walk, Green Function, and Equilibrium Measure
Authors : Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov
Published in: An Introduction to Random Interlacements
Publisher: Springer International Publishing
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Abstract
In this chapter we collect some preliminary facts that we will need in the sequel. In Sect. 1.1 we introduce our basic notation related to subsets of \({\mathbb{Z}}^{d}\) and functions on \({\mathbb{Z}}^{d}\). In Sect. 1.2 we introduce simple random walk on \({\mathbb{Z}}^{d}\) and discuss properties of the Green function in the transient case d ≥ 3. In Sect. 1.3 we discuss the basics of potential theory, introduce the notion of equilibrium measure and capacity, and derive some of their properties.