PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

250 Geoffrey Grimmett
now make two comparisons, the effect of each of which is to increase all
effective resistances of the network. At the first stage, we replace all finite
edge-resistances l(e) −1 by unit resistances. This cannot decrease any effective
resistance. At the next stage we replace each rw point by
+ with probability π(+; α, β) − p+
ρ (�= +) with probability π(ρ; α, β) − (1 − prw − p+)π(ρ),
in accordance with (10.20) and (10.21) (and where α, β are chosen so that
(10.20), (10.21) hold, and furthermore βc(α, 2) < β < 1). Such replacements
can only increase effective resistance.
In this way we obtain a comparison between the resistance of the network
arising from the above labyrinth and that of the labyrinth L(ω) defined
around (10.19). Indeed, it suffices to prove that the effective resistance between
0 and ∞ (in the infinite equivalence class) of the labyrinth L(ω) is a.s.
finite. By examining the geometry, we claim that this resistance is no greater
(up to a multiplicative constant) than the resistance between the origin and
infinity of the corresponding infinite open percolation cluster of ω. By the
next lemma, the last resistance is a.s. finite, whence the original walk is
a.s. transient (when confined to the almost surely unique infinite equivalence
class).
Lemma 10.22. Let d ≥ 3, 0 < α < 1, and βc(α, 2) < β < 1. Let R be the
effective resistance between the origin and the points at infinity, in the above
bond percolation process ω on L. Then
� � �
R < ∞ �0 belongs to the infinite open cluster = 1.
Pα,β
Presumably the same conclusion is valid under the weaker hypothesis that
β > βc(α, d).
Sketch Proof. Rather than present all the details, here are some notes. The
main techniques used in [163] arise from [164], and principally one uses the
exponential decay noted in Lemma 10.4. That such decay is valid whenever
β > βc(α, d) uses the machinery of [164]. This machinery may be developed in
the present setting (in [164] it is developed only for the hypercubic lattice L d ).
Alternatively, ‘slab arguments’ show (a) and (b) of Lemma 10.4 for sufficiently
large β; certainly the condition β > βc(α, 2) suffices for the conclusion. �
Comments on the Proof of Lemma 10.16. This resembles closely the proof
of the uniqueness of the infinite percolation cluster (Theorem 7.1). We do
not give the details. The notion of ‘trifurcation’ is replaced by that of an
‘encounter zone’. Let R ≥ 1 and B = B(R). A translate x + B is called an
encounter zone if
(a) all points in x + B are rw points, and
(b) in the labyrinth Z d \{x+B}, there are three or more infinite equivalence
classes which are part of the same equivalence class of L d .
Note that different encounter zones may overlap. See [73] for details. �