PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

Percolation and Disordered Systems 249
satisfies π(ρ; α, β) > 0 if 0 < α, β < 1, and furthermore
Also,
π(+; α, β) = β d (1 − α) (2d
2 )−d .
π(∅; α, β) = Pα,β(ρ0 = ∅) = 1 − �
π(ρ; α, β).
Let prw, p+ satisfy prw, p+ > 0, prw + p+ ≤ 1. We pick α, β such that
0 < α, β < 1, β > βc(α, 2) and
ρ∈R
(10.20) π(+; α, β) ≥ 1 − prw (≥ p+).
(That this may be done is a consequence of the fact that βc(α, 2) < 1 for all
α > 0; cf. Lemma 10.18.)
With this choice of α, β, let
� �
π(ρ; α, β)
A = min : ρ �= +, ρ ∈ R
π(ρ)
with the convention that 1/0 = ∞. (Thus defined, A depends on π as well as
on prw. If we set A = min � π(ρ; α, β) : ρ �= +, ρ ∈ R � , we obtain a (smaller)
constant which is independent of π, and we may work with this definition
instead.) Then
π(ρ; α, β) ≥ Aπ(ρ) for all ρ �= +,
and in particular
(10.21) π(ρ; α, β) ≥ (1 − prw − p+)π(ρ) if ρ �= +
so long as p+ satisfies 1 − prw − p+ < A. Note that A = A(α, β) > 0.
We have from the fact that β > βc(α, 2) that the percolation process ω
a.s. contains an infinite open cluster. It follows that there exists a.s. a rw
point in L(ω) which is L(ω)-non-localised. The labyrinth Z of the theorem
may be obtained (in distribution) from L as follows. Having sampled L(ω),
we replace any crossing (resp. reflector ρ (�= +)) by a rw point with probability
π(+; α, β) − p+ (resp. π(ρ; α, β) − (1 − prw − p+)π(ρ)); cf. (10.20) and
(10.21). The ensuing labyrinth L ′ (ω) has the same probability distribution
as Z. Furthermore, if L(ω) is non-localised, then so is L ′ (ω).
The first part of Theorem 10.17 has therefore been proved. Assume henceforth
that d ≥ 3, and consider part (b).
Now consider a labyrinth defined by prw, p+, π(·). Let e be an edge of Z d .
Either e lies in a unique path joining two rw points (but no other rw point)
of some length l(e), or it does not (in which case we set l(e) = 0). Now,
the random walk in this labyrinth induces an embedded Markov chain on the
set of rw points. This chain corresponds to an electrical network obtained
by placing an electrical resistor at each edge e having resistance l(e) −1 . We