PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

240 Geoffrey Grimmett
p+
1
pc(site)
Fig. 10.5. The grey region and the heavy lines of the figure indicate the part of
(prw, p+) space for which non-localisation is proved. The labyrinth is a.s. localised
when prw = p+ = 0; see Theorem 10.9.
Theorem 10.11. Let prw > 0. There exists a strictly positive constant
A = A(prw) such that the following holds. The labyrinth Z is P-a.s. nonlocalised
if any of the following conditions hold:
(a) prw > pc(site), the critical probability of site percolation on L 2 ,
(b) p+ = 0,
(c) prw + p+ > 1 − A.
We shall see in the proof of part (c) (see Theorem 10.17) that A(prw) → 0
as prw ↓ 0. This fact is reported in Figure 10.5, thereby correcting an error
in the corresponding figure contained in [165].
Proof of Theorem 10.10. Assume prw > 0. We shall compare the labyrinth
with a certain electrical network. By showing that the effective resistance
of this network between 0 and ∞ is a.s. infinite, we shall deduce that Z is
a.s. recurrent. For details of the relationship between Markov chains and
electrical networks, see the book [117] and the papers [252, 276].
By the term Z-path we mean a path of the lattice (possibly with selfintersections)
which may be followed by the light; i.e., at rw points it is
unconstrained, while at reflectors and crossings it conforms to the appropriate
rule. A formal definition will be presented in Section 10.3.
Let e = 〈u, v〉 be an edge of L 2 . We call e a normal edge if it lies in
some Z-path π(e) which is minimal with respect to the property that its two
endvertices (and no others) are rw points, and furthermore that these two
endvertices are distinct. If e is normal, we write l(e) for the number of edges
in π(e); if e is not normal, we write l(e) = 0. We define ρ(e) = 1/l(e), with
the convention that 1/0 = ∞.
1
prw