PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

238 Geoffrey Grimmett
We now use the above ‘labyrinth’ to define a bond percolation process on
L2 A . We declare the edge of L2 1 1 1 1
A joining (m − 2 , n − 2 ) to (m+ 2 , n+ 2 ) to be
open if there is a NE mirror at (m, n); similarly we declare the edge joining
(m − 1 1 1 1
2 , n+ 2 ) to (m+ 2 , n − 2 ) to be open if there is a NW mirror at (m, n).
Edges which are not open are designated closed. This defines a percolation
model in which north-easterly edges (resp. north-westerly edges) are open
with probability pNE = 1
2 (resp. pNW = 1
2 ). Note that pNE + pNW = 1, which
implies that the percolation model is critical (see [G, 203]).
Let N be the number of open circuits in L2 A which contain the origin
in their interiors. Using general results from percolation theory, we have
that P(N ≥ 1) = 1, where P is the appropriate probability measure. (This
follows from the fact that θ( 1
2 ) = 0; cf. Theorem 9.1, see also [G, 182, 203].)
However, such an open circuit corresponds to a barrier of mirrors surrounding
the origin, from which no light can escape (see Figure 10.4 again). Therefore
η(1) = 1.
We note that the above proof is valid in the slightly more general setting
in which NE mirrors are present with density pNE and NW mirrors with
density pNW where pNW + pNE = 1 and 0 < pNW < 1. This generalisation
was noted in [81]. �
When 0 < p < 1, the question of whether or not η(p) = 1 is wide open,
despite many attempts to answer it 7 . It has been conjectured that η(p) = 1
for all p > 0, based on numerical simulations; see [110, 375]. Some progress
has been made recently by Quas [320].
The above lattice version of the ‘mirror model’ appears to have been
formulated first around 20 years ago. In a systematic approach to random
environments of reflectors, Ruijgrok and Cohen [324] proposed a programme
of study of ‘mirror’ and ‘rotator’ models. Since then, there have been reports
of many Monte Carlo experiments, and several interesting conjectures have
emerged (see [108, 109, 110, 343, 375]). Rigorous progress has been relatively
slight; see [81, G, 320] for partial results.
The principal difficulty in the above model resides in the facts that the
environment is random but that the trajectory of the light is (conditionally)
deterministic. If we relax the latter determinism, then we arrive at model
which is more tractable. In this new version, there are exactly three types of
point, called mirrors, crossings, and random walk (rw) points. Let prw, p+ ≥
0 be such that prw + p+ ≤ 1. We designate each vertex x to be
a random walk (rw) point, with probability prw,
a crossing, with probability p+,
a mirror, otherwise.
If a vertex is a mirror, then it is occupied by a NW reflector with probability
, and otherwise by a NE reflector. The environment of mirrors and rw points
1
2
7 I heard of this problem in a conversation with Hermann Rost and Frank Spitzer in
Heidelberg in 1978. The proof that η(1) = 1 was known to me (and presumably to others)
in 1978 also.