PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

194 Geoffrey Grimmett
7.2 Percolation in Slabs
Many results were proved for subcritical percolation under the hypothesis
of ‘finite susceptibility’, i.e., that χ(p) = Ep|C| satisfies χ(p) < ∞. Subsequently,
it was proved in [12, 267, 270] that this hypothesis is satisfied
whenever p < pc. The situation was similar for supercritical percolation, the
corresponding hypothesis being that percolation occurs in slabs. We define
the slab of thickness k by
Sk = Z d−1 × {0, 1, . . . , k},
with critical probability pc(Sk); we assume here that d ≥ 3. The decreasing
limit pc(S) = limk→∞ pc(Sk) exists, and satisfies pc(S) ≥ pc. The hypothesis
of ‘percolation in slabs’ is that p > pc(S). Here is an example of the
hypothesis in action (cf. Theorem 6.2 and equation (6.8)).
Theorem 7.6. The limit
�
σ(p) = lim −
n→∞
1
n log Pp(0
�
↔ ∂B(n), |C| < ∞)
exists. Furthermore σ(p) > 0 if p > pc(S).
This theorem asserts the exponential decay of a ‘truncated’ connectivity
function when d ≥ 3. Corresponding results when d = 2 may be proved using
duality.
Proof. The existence of the limit is an exercise in subadditivity (see [94, G]),
and we sketch here only a proof that σ(p) > 0. Assume that p > pc(S), so
that p > pc(Sk) for some k; choose k accordingly. Let Hn be the hyperplane
containing all vertices x with x1 = n. It suffices to prove that
(7.7) Pp(0 ↔ Hn, |C| < ∞) ≤ e −γn
for some γ = γ(p) > 0. Define the slabs
Ti = {x ∈ Z d : (i − 1)k ≤ x1 < ik}, 1 ≤ i < ⌊n/k⌋.
Any path from 0 to Hn must traverse every such slab. Since p > pc(Sk), each
slab a.s. contains an infinite open cluster. If 0 ↔ Hn and |C| < ∞, then all
paths from 0 to Hn must evade all such clusters. There are ⌊n/k⌋ slabs to
traverse, and a price is paid for each. With a touch of rigour, this argument
implies that
Pp(0 ↔ Hn, |C| < ∞) ≤ {1 − θk(p)} ⌊n/k⌋
where
θk(p) = Pp(0 ↔ ∞ in Sk) > 0.
For more details, see [G]. �
Grimmett and Marstrand [164] proved that pc = pc(S), using ideas similar
to those of [48, 49]. This was achieved via a ‘block construction’ which
appears to be central to a full understanding of supercritical percolation and
to have further applications elsewhere. The details are presented next.