PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

220 Geoffrey Grimmett
where α ′ = α ′ (p) is strictly positive and continuous for 0 < p < 1 (and we
have used the fact (6.7) that χ(pc) = ∞). �
Finally we discuss the verification of the triangle condition T(pc) < ∞.
This has been proved for large d (currently d ≥ 19) by Hara and Slade [177,
178, 179, 180, 181], and is believed to hold for d ≥ 7. The corresponding
condition for a ‘spread-out’ percolation model, having large but finite-range
links rather than nearest-neighbour only, is known to hold for d > 6.
The proof that T(pc) < ∞ is long and technical, and is to be found in
[179]; since the present author has no significant improvement on that version,
the details are not given here. Instead, we survey briefly the structure of the
proof.
The triangle function (8.9) involves convolutions, and it is therefore natural
to introduce the Fourier transform of the connectivity function τp(x, y) =
Pp(x ↔ y). More generally, if f : Z d → R is summable, we define
�f(θ) = �
f(x)e iθ·x , for θ = (θ1, . . . , θd) ∈ [−π, π] d ,
x∈Z d
where θ · x = �d j=1 θjxj. If f is symmetric (i.e., f(x) = f(−x) for all x),
then � f is real.
We have now that
(8.23) T(p) = �
τp(0, x)τp(x, y)τp(y, 0) = (2π) −d
�
�τp(θ) 3 dθ.
x,y
[−π,π] d
The proof that T(pc) < ∞ involves an upper bound on �τp, namely the so
called infra-red bound
(8.24) �τp(θ) ≤ c(p)
|θ| 2
where |θ| = √ θ · θ. It is immediate via (8.23) that the infra-red bound (8.24)
implies that T(p) < ∞. Also, if (8.24) holds for some c(p) which is uniformly
bounded for p < pc, then T(pc) = limp↑pc T(p) < ∞.
It is believed that
(8.25) �τp(θ) ≃ 1
|θ| 2−η as |θ| → 0
where η is the critical exponent given in the table of Section 8.2.
Theorem 8.26 (Hara–Slade [179]). There exists D satisfying D > 6 such
that, if d ≥ D, then
�τp(θ) ≤ c(p)
|θ| 2