PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

Percolation and Disordered Systems 243
We now define bond percolation processes on L2 A and L2B . Assume p+ = 0.
We present the rules for L2 A only; the rules for L2B are analogous. An edge of
L2 1 1 1 1
A joining (m − 2 , n − 2 ) to (m + 2 , n + 2 ) to declared to be open if there is
a NE mirror at (m, n); similarly we declare the edge joining (m − 1 1
2 , n + 2 )
to (m + 1 1
2 , n − 2 ) to be open if there is a NW mirror at (m, n). Edges which
are not open are called closed. This defines percolation models on L2 A and
L2 B in which north-easterly edges (resp. north-westerly edges) are open with
probability pNE = 1
2 (1 − prw) (resp. pNW = 1
2 (1 − prw)). These processes are
subcritical since pNE + pNW = 1 − prw < 1. Therefore, there exists (P-a.s.)
no infinite open path in either L 2 A or L2 B
, and we assume henceforth that no
such infinite open path exists.
Let N(A) (resp. N(B)) be the number of open circuits in L 2 A (resp. L2 B )
which contain the origin in their interiors. Since the above percolation processes
are subcritical, there exists (by Theorem 6.10) a strictly positive constant
α = α(pNW, pNE) such that
(10.14) �
P x lies in an open cluster of L 2 A
�
of diameter at least n ≤ e −αn
for all n,
for any vertex x of L2 A . (By the diameter of a set C of vertices, we mean
max{|y −z| : y, z ∈ C}.) The same conclusion is valid for L2 B . We claim that
(10.15) P � 0 is a rw point, and N(A) = N(B) = 0 � > 0,
and we prove this as follows. Let Λ(k) = [−k, k] 2 , and let Nk(A) (resp.
Nk(B)) be the number of circuits contributing to N(A) (resp. N(B)) which
contain only points lying strictly outside Λ(k). If Nk(A) ≥ 1 then there exists
some vertex (m + 1 1
2 , 2 ) of L2A , with m ≥ k, which belongs to an open circuit
of diameter exceeding m. Using (10.14),
P � Nk(A) ≥ 1 � ≤
∞�
m=k
e −αm < 1
3
for sufficiently large k. We pick k accordingly, whence
P � Nk(A) + Nk(B) ≥ 1 � ≤ 2
3 .
Now, if Nk(A) = Nk(B) = 0, and in addition all points of L2 inside Λ(k) are
rw points, then N(A) = N(B) = 0. These last events have strictly positive
probabilities, and (10.15) follows.
Let J be the event that there exists a rw point x = x(Z) which lies in
the interior of no open circuit of either L 2 A or L2 B
. Since J is invariant with
respect to translations of L 2 , and since P is product measure, we have that
P(J) equals either 0 or 1. Using (10.15), we deduce that P(J) = 1. Therefore
we may find a.s. some such vertex x = x(Z). We claim that x is Z-nonlocalised,
which will imply as claimed that the labyrinth if a.s. non-localised.