PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

246 Geoffrey Grimmett
A L d -path is defined to be a sequence x0, e0, x1, e1, . . . of alternating vertices
xi and distinct edges ej such that ej = 〈xj, xj+1〉 for all j. If the path
has a final vertex xn, then it is said to have length n and to join x0 to xn.
If it is infinite, then it is said to join x0 to ∞. A L d -path may visit vertices
more than once, but we insist that its edges be distinct.
We define a Z-path to be a L d -path x0, e0, x1, e1, . . . with the property
that, for all j,
xj+1 − xj = Zxj(xj − xj−1) whenever Zxj �= ∅,
which is to say that the path conforms to all reflectors.
Let N be the set of rw points. We define an equivalence relation ↔ on N
by x ↔ y if and only if there exists a Z-path with endpoints x and y. We
denote by Cx the equivalence class of (N, ↔) containing the rw point x, and
by C the set of equivalence classes of (N, ↔). The following lemma will be
useful; a sketch proof is deferred to the end of the section.
Lemma 10.16. Let d ≥ 2 and prw > 0. The number M of equivalence
classes of (N, ↔) having infinite cardinality satisfies
either P(M = 0) = 1 or P(M = 1) = 1.
Next we define a random walk in the random labyrinth Z. Let x be a rw
point. A walker, starting at x, flips a fair coin (in the manner of a symmetric
random walker) whenever it arrives at a rw point in order to determine its
next move. When it meets a reflector, it moves according to the reflector (i.e.,
if it strikes the reflector ρ in the direction u, then it departs in the direction
ρ(u)). Writing P Z x
for the law of the walk, we say that the point x is Z-
recurrent if P Z x (XN = x for some N ≥ 1) = 1, and Z-transient otherwise.
As before, we say that Z is transient if there exists a rw point x which is
Z-transient, and recurrent otherwise.
Note that, if the random walker starts at the rw point x, then the sequence
of rw points visited constitutes an irreducible Markov chain on the
equivalence class Cx. Therefore, the rw point x is Z-localised if and only if
|Cx| < ∞. As before, we say that Z is localised if all rw points are Z-localised,
and non-localised otherwise.
Theorem 10.17. Let prw > 0. There exists a strictly positive constant
A = A(prw, d) such that the following holds.
(a) Assume that d ≥ 2. If 1 −prw −p+ < A, then the labyrinth Z is P-a.s.
non-localised.
(b) Assume that d ≥ 3. If 1 − prw − p+ < A, then Z is P-a.s. transient.
As observed after 10.11, we have that A = A(prw, d) → 0 as prw ↓ 0.
Using methods presented in [114, 115, 220], one may obtain an invariance
principle for a random walk in a random labyrinth, under the condition that
1 − prw − p+ is sufficiently small. Such a principle is valid for a walk which