PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

248 Geoffrey Grimmett
where pc(site) is the critical probability of site percolation on L2 . Now tile
Z2 with copies of B(L + 1
2 ), overlapping at the sides. It follows from the
obvious relationship with site percolation that, with positive probability, the
origin lies in an infinite cluster. �
We now construct a labyrinth on Z d from each realisation ω ∈ {0, 1} F of
the percolation process (where we write ω(f) = 1 if and only if the edge f
is open). That is, with each point x ∈ Z d we shall associate a member ρx
of R ∪ {∅}, in such a way that ρx depends only on the edges 〈e1, e2〉 of F
for which e1 and e2 are distinct edges of L d having common vertex x. It
will follow that the collection {ρx : x ∈ Z d } is a family of independent and
identically distributed objects.
Since we shall define the ρx according to a translation-invariant rule, it will
suffice to present only the definition of the reflector ρ0 at the origin. Let E0 be
the set of edges of L d which are incident to the origin. There is a natural one–
one correspondence between E0 and I ± , namely, the edge 〈0, u〉 corresponds
to the unit vector u ∈ I ± . Let ρ ∈ R. Using the above correspondence, we
may associate with ρ a set of configurations in Ω = {0, 1} F , as follows. Let
Ω(ρ, 0) be the subset of Ω containing all configurations ω satisfying
ω � 〈0, u1〉, 〈0, u2〉 � = 1 if and only if ρ(−u1) = u2
for all distinct pairs u1, u2 ∈ I ± .
It is not difficult to see that
Ω(ρ1, 0) ∩ Ω(ρ2, 0) = ∅ if ρ1, ρ2 ∈ R, ρ1 �= ρ2.
Let ω ∈ Ω be a percolation configuration on F. We define the reflector
ρ0 = ρ0(ω) at the origin by
�
ρ if ω ∈ Ω(ρ, 0)
(10.19) ρ0 =
∅ if ω /∈ �
ρ∈R Ω(ρ, 0).
If ρ0 = ρ ∈ R, then the behaviour of a light beam striking the origin
behaves as in the corresponding percolation picture, in the following sense.
Suppose light is incident in the direction u1. There exists at most one direction
u2 (�= u1) such that ω(〈0, −u1〉, 〈0, u2〉) = 1. If such a u2 exists, then
the light is reflected in this direction. If no such u2 exists, then it is reflected
back on itself, i.e., in the direction −u1.
For ω ∈ {0, 1} F , the above construction results in a random labyrinth
L(ω). If the percolation process contains an infinite open cluster, then the
corresponding labyrinth contains (a.s.) an infinite equivalence class.
Turning to probabilities, it is easy to see that, for ρ ∈ R,
π(ρ; α, β) = Pα,β(ρ0 = ρ)