PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

Percolation and Disordered Systems 247
starts in the (a.s) unique infinite equivalence class of Z. The details will
appear in [73].
The following proof of Theorem 10.17 differs from that presented in [165] 8 .
It is slightly more complicated, but gives possibly a better numerical value
for the constant A.
Proof. The idea is to relate the labyrinth to a certain percolation process,
as follows. We begin with the usual lattice L d = (Z d , E d ), and from this we
construct the ‘line lattice’ (or ‘covering lattice’) L as follows. The vertex set
of L is the edge-set E d of L d , and two distinct vertices e1, e2 (∈ E d ) of L are
called adjacent in L if and only if they have a common vertex of L d . If this
holds, we write 〈e1, e2〉 for the corresponding edge of L, and denote by F the
set of all such edges. We shall work with the graph L = (E d , F), and shall
construct a bond percolation process on L. We may identify E d with the set
of midpoints of members of E; this embedding is useful in visualising L.
Let 〈e1, e2〉 ∈ E d . If the edges e1 and e2 of L d are perpendicular, we
colour 〈e1, e2〉 amber, and if they are parallel blue. Let 0 ≤ α, β ≤ 1. We
declare an edge 〈e1, e2〉 of F to be open with probability α (if amber) or β
(if blue). This we do for each 〈e, f〉 ∈ F independently of all other members
of F. Write Pα,β for the corresponding probability measure, and let θ(α, β)
be the probability that a given vertex e (∈ E d ) of L is in an infinite open
cluster of the ensuing percolation process on L. It is easily seen that θ(α, β)
is independent of the choice of e.
Lemma 10.18. Let d ≥ 2 and 0 < α ≤ 1. then
satisfies βc(α, d) < 1.
βc(α, d) = sup{β : θ(α, β) = 0}
Note that, when d = 2 and β = 0, the process is isomorphic to bond percolation
on L 2 with edge-parameter α. Therefore θ(α, 0) > 0 and βc(α, 2) = 0
when α > 1
2 .
Proof. Since βc(α, d) is non-increasing in d, it suffices to prove the conclusion
when d = 2. Henceforth assume that d = 2.
Here is a sketch proof. Let L ≥ 1 and let AL be the event that every
vertex of L lying within the box B(L + 1
2
) = [−L − 1
2
, L + 1
2 ]2 (of R 2 ) is
joined to every other vertex lying within B(L + 1
2 ) by open paths of L lying
inside B(L + 1
2 ) which do not use boundary edges. For a given α satisfying
0 < α < 1, there exist L and β ′ such that
Pα,β ′(AL) ≥ pc(site),
8 There is a small error in the proof of Theorem 7 of [165], but this may easily be
corrected.