PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

234 Geoffrey Grimmett
the volume of ∂B(β k−1 ) has order β (k−1)(d−1) . The above construction can
therefore only succeed when 4 k < β (k−1)(d−1) for all large k, which is to say
that β > 4 1/(d−1) .
This crude picture suggests the necessary inequalities
(10.3) 4 1/(d−1) < β < 4,
which can be satisfied if and only if d ≥ 3.
Assume now that d ≥ 3. Our target is to show that the infinite cluster I
contains sufficiently many disjoint paths to enable a comparison of its effective
resistance with that of the tree in Figure 10.1, and with some value of β
satisfying (10.3). In presenting a full proof of this, we shall use the following
two percolation estimates, which are consequences of the results of Chapter
7.
Lemma 10.4. Assume that p > pc.
(a) There exists a strictly positive constant γ = γ(p) such that
� � −γn
(10.5) Pp B(n) ↔ ∞ ≥ 1 − e for all n.
(b) Let σ > 1, and let A(n, σ) be the event that there exist two vertices inside
B(n) with the property that each is joined by an open path to ∂B(σn) but
that there is no open path of B(σn) joining these two vertices. There exists
a strictly positive constant δ = δ(p) such that
� � −δn(σ−1)
(10.6) Pp A(n, σ) ≤ e
for all n.
We restrict ourselves here to the case d = 3; the general case d ≥ 3 is
similar. The surface of B(n) is the union of six faces, and we concentrate on
the face
F(n) = {x ∈ Z 3 : x1 = n, |x2|, |x3| ≤ n}.
We write Bk = B(3 k ) and Fk = F(3 k ). On Fk, we distinguish 4 k points,
namely
xk(i, j) = (idk, jdk), −2 k−1 < i, j ≤ 2 k−1
where dk = ⌊(4/3) k ⌋. The xk(i, j) are distributed on Fk in the manner of a
rectangular grid, and they form the ‘centres of attraction’ corresponding to
the kth generation of the tree discussed above.
With each xk(i, j) we associate four points on Fk+1, namely those in the
set
Ik(i, j) = {xk+1(r, s) : r = 2i − 1, 2i, s = 2j − 1, 2j}.
These four points are called children of xk(i, j). The centroid of Ik(i, j) is
denoted Ik(i, j). We shall attempt to construct open paths from points near
xk(i, j) to points near to each member of Ik(i, j), and this will be achieved
with high probability. In order to control the geometry of such paths, we
shall build them within certain ‘tubes’ to be defined next.