PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

Percolation and Disordered Systems 237
NW NE
Fig. 10.3. NW and NE reflectors in action.
Fig. 10.4. (a) The heavy lines form the lattice L2 A , and the central point is the origin
of L2 . (b) An open circuit in L2 A constitutes a barrier of mirrors through which no
light may penetrate.
place nothing at x. This is done independently for different x. If a reflector
is placed at x, then we specify that it is equally likely to be NW as NE.
We shine a torch northwards from the origin. The light is reflected by
the mirrors, and we ask whether or not the light ray returns to the origin.
Letting
η(p) = Pp(the light ray returns to the origin),
we would like to know for which values of p it is the case that η(p) = 1. It
is reasonable to conjecture that η is non-decreasing in p. Certainly η(0) = 0,
and it is ‘well known’ that η(1) = 1.
Theorem 10.9. It is the case the η(1) = 1.
Proof of Theorem 10.9. This proof is alluded to in [G] and included in [81].
From L2 we construct an ancillary lattice L2 A as follows. Let
�
A =
(m + 1
2
�
1 , n + 2 ) : m + n is even .
On A we define the adjacency relation ∼ by (m + 1 1 1 1
2 , n + 2 ) ∼ (r + 2 , s + 2 )
if and only if |m − r| = |n − s| = 1, obtaining thereby a copy of L2 denoted
as L2 A . See Figure 10.4.