PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

Percolation and Disordered Systems 245
there exists (by assumption) no infinite open path, this proves that f lies in
an open circuit of F in L2 A .
By taking the union over all e ∈ F, we obtain that F is a union of open
circuits of L 2 A and L2 B
. Each such circuit has an interior and an exterior, and
x lies (by assumption, above) in every exterior. There are various ways of
deducing that x is Z-non-localised, and here is such a way.
Assume that x is Z-localised. Amongst the set of vertices {x + (n, 0) :
n ≥ 1}, let y be the rightmost vertex at which there lies a frontier mirror.
By the above argument, y belongs to some open circuit G of F (belonging to
either L 2 A or L2 B
), whose exterior contains x. Since y is rightmost, we have
that y ′ = y + (−1, 0) is illuminated by light originating at x, and that light
traverses the edge 〈y ′ , y〉. Similarly, light does not traverse the edge 〈y, y ′′ 〉,
where y ′′ = y +(1, 0). Therefore, the point y +( 1
2 , 0) of R2 lies in the interior
of G, which contradicts the fact that y is rightmost. This completes the proof
for part (b). �
10.3 General Labyrinths
There are many possible types of reflector, especially in three and more dimensions.
Consider Z d where d ≥ 2. Let I = {u1, u2, . . . , ud} be the set of
positive unit vectors, and let I ± = {−1, +1} × I; members of I ± are written
as ±uj. We make the following definition. A reflector is a map ρ : I ± → I ±
satisfying ρ(−ρ(u)) = −u for all u ∈ I ± . We denote by R the set of reflectors.
The ‘identity reflector’ is called a crossing (this is the identity map on I ± ),
and denoted by +.
The physical interpretation of a reflector is as follows. If light impinges
on a reflector ρ, moving in a direction u (∈ I ± ) then it is required to depart
the reflector in the direction ρ(u). The condition ρ(−ρ(u)) = −u arises from
the reversibility of reflections.
Using elementary combinatorics, one may calculate that the number of
distinct reflectors in d dimensions is
d�
s=0
(2d)!
(2s)! 2 d−s (d − s)! .
A random labyrinth is constructed as follows. Let prw and p+ be nonnegative
reals satisfying prw +p+ ≤ 1. Let Z = (Zx : x ∈ Zd ) be independent
random variables taking values in R ∪ {∅}, with common mass function
⎧
⎪⎨ prw
if α = ∅
P(Z0 = α) = p+
if α = +
⎪⎩
(1 − prw − p+)π(ρ) if α = ρ ∈ R\{+},
where π is a prescribed probability mass function on R\{+}. We call a point
x a crossing if Zx = +, and a random walk (rw) point if Zx = ∅.