An Introduction to Random Interlacements by Alexander Drewitz Balázs Ráth & Artëm Sapozhnikov

Author:Alexander Drewitz, Balázs Ráth & Artëm Sapozhnikov
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

6.3 Proof of Proposition 6.3

We begin with a generating function calculation. For w ∈ W +, let

denote the number of visits of w to F. Let

(6.3.1)

where is the law of a (d − 2)-dimensional simple random walk started from the origin, is the first time when this random walk returns to 0, cf. (1.​2.​3), and g d−2 is the Green function for this random walk.

Note that it follows from Pólya’s theorem (cf. Theorem 1.7) that q > 0 if and only if d ≥ 5.

Lemma 6.4.

If λ > 0 and d ≥ 5 satisfy

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