1347–1356. MR1733151 [2] Benjamini, I. and Schramm, O. 1996. Percolation beyond Z Paper no. 8, 71–82. MR1423907 97j:60179 487–507. MR1815220 2002h:82049 205–221. MR1772346 2001i:82036 1019–1039. MR2031859 2005c:82051
This proves Equation 28.
We are now ready to complete the proof of Theorem 4.
Proof of Theorem 4 for β and δ: We shall verify the hypotheses of Lemma 12. Similarly to the
proof of Theorem 3, let H
1
, H
2
, and H
3
be three lattice-half-spaces whose neighbourhoods are pairwise disjoint [i.e.
N H
j
∩ N H
k
= ; for 1 ≤ j k ≤ 3]. Let u
1
, u
2
and u
3
be vertices such that, for each i, u
i
is contained in the boundary plane of H
i
. For y
∈ H
1
, we define the event E
1
[ y] = { y ∈ C u
1
, C u
1
⊂ H
1
, u
1
6↔ Q }. For i = 2, 3, define the events
E
i
= { C u
i
⊂ H
i
, u
i
↔ Q }. Then, for any y
∈ H
1
, the three events E
1
[ y], E
2
, and E
3
are independent, since they depend only on events in
N H
1
, N H
2
, and N H
3
respectively. For p and q as in Lemma 14, we have X
y ∈V
P
p,q
{ u
1
↔ y, u
1
6↔ Q, u
2
↔ Q, u
3
↔ Q, u
2
6↔ u
3
} ≥
X
y ∈H
1
P
p,q
E
1
[ y] ∩ E
2
∩ E
3
= X
y ∈H
1
P
p,q
E
1
[ y] P
p,q
E
2
P
p,q
E
3
≥ ε
Q
χp, q ε
θ
θ p, q
2
[by Lemma 14]. This proves 25, and hence we have shown that
β = 1 and δ = 2.
Acknowledgements
We are grateful to Karoly Bezdek, Ragnar-Olaf Buchweitz, Asia Weiss, and Walter Whiteley for helpful discussions about hyperbolic geometry, and to a referee for helpful comments.
References
[1] Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. 1999. Critical percolation on any nona- menable group has no infinite clusters. Ann. Probab.
27, 1347–1356. MR1733151 [2] Benjamini, I. and Schramm, O. 1996. Percolation beyond Z
d
, many questions and a few answers. Electronic Commun. Probab.