PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

Percolation and Disordered Systems 239
is denoted by Z = (Zx : x ∈ Z 2 ) and is termed a ‘labyrinth’; we write P for
the probability measure associated with the labyrinth, so that P is product
measure on the corresponding environment space.
The physical meaning of these terms is as follows. Suppose that some
vertex x is occupied by a candle, which emits light rays along the four axes
leaving x. When a ray is incident with a mirror, then it is reflected accordingly.
When a ray encounters a crossing, then it continues undeflected.
When a ray encounters a rw point, then it leaves this point in one of the four
available directions, chosen at random in the manner of a random walk.
We formalise this physical explanation by defining a type of random walk
X = (X0, X1, . . . ) on L 2 . Assume that prw > 0, and sample a random
labyrinth Z according to the measure P. Let x be a rw point, and set X0 = x.
We choose a random neighbour X1 of x, each of the four possibilities being
equally likely. Having constructed X0, X1, . . .,Xr, we define Xr+1 as follows.
If Xr is a rw point, we let Xr+1 be a randomly chosen neighbour of Xr (chosen
independently of all earlier choices); if Xr is not a rw point, then we define
Xr+1 to be the next vertex illuminated by a ray of light which is incident
with Xr travelling in the direction Xr −Xr−1. The consequent sequence X is
called a ‘random walk in a random labyrinth’. Let P Z x denote the law of X,
conditional on Z, and starting at x. We say that the rw point x is Z-recurrent
if there exists (P Z x -a.s.) an integer N such that XN = x, and otherwise we
say that x is Z-transient. We say that the labyrinth Z is recurrent if every
rw point is Z-recurrent. It is easily seen, using the translation-invariance of
P and the zero–one law, that the labyrinth is P-a.s. recurrent if and only if
P � 0 is Z-recurrent � � 0 is a rw point � = 1.
Theorem 10.10. If prw > 0 then the labyrinth Z is P-a.s. recurrent.
This theorem, together with most other results in this chapter, appears in
[165], and is proved by showing that a corresponding electrical network has
infinite resistance. A brief proof appears at the end of this section.
Remembering that irreducible Markov chains on finite state spaces are
necessarily recurrent, we turn our attention to a question of ‘localisation’.
Let x be a rw point in the random labyrinth Z, and let X be constructed
as above. We say that x is Z-localised if X visits (P Z x -a.s.) only finitely
many vertices; we call x Z-non-localised otherwise. We say that the random
labyrinth Z is localised if all rw points are Z-localised, and we call it nonlocalised
otherwise. Using the translation-invariance of Z and the zero–one
law, we may see that Z is P-a.s. localised if and only if
P � 0 is Z-localised � � 0 is a rw point � = 1.