PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

Percolation and Disordered Systems 215
Turning to percolation on L d , it is known that the critical exponents agree
with those of a regular tree when d is sufficiently large. In fact, this is believed
to hold if and only if d ≥ 6, but progress so far assumes that d ≥ 19. In
the following theorem, taken from [180], we write f(x) ≃ g(x) if there exist
positive constants c1, c2 such that c1f(x) ≤ g(x) ≤ c2f(x) for all x close to a
limiting value.
Theorem 8.4. If d ≥ 19 then
(8.5)
(8.6)
(8.7)
(8.8)
θ(p) ≃ (p − pc) 1
χ(p) ≃ (pc − p) −1
as p ↓ pc,
as p ↑ pc,
1 −
ξ(p) ≃ (pc − p) 2 as p ↑ pc,
χ f k+1 (p)
χ f k (p) ≃ (pc − p) −2
as p ↑ pc, for k ≥ 1.
Note the strong form of the asymptotic relation ≃, and the identification
of the critical exponents β, γ, ∆, ν. The proof of Theorem 8.4 centres on a
property known as the ‘triangle condition’. Define
(8.9) T(p) = �
x,y∈Z d
and introduce the following condition,
(8.10) Triangle condition: T(pc) < ∞.
Pp(0 ↔ x)Pp(x ↔ y)Pp(y ↔ 0),
The triangle condition was introduced by Aizenman and Newman [26], who
showed that it implied that χ(p) ≃ (pc − p) −1 as p ↑ pc. Subsequently
other authors showed that the triangle condition implied similar asymptotics
for other quantities. It was Hara and Slade [179] who verified the triangle
condition for large d, exploiting a technique known as the ‘lace expansion’.
We present no full proof of Theorem 8.4 here, pleading two reasons. First,
such a proof would be long and complicated. Secondly, we are unable to do
better than is already contained in the existing literature (see [179, 180]).
Instead, we (nearly) prove the above Aizenman–Newman result (equation
(8.6) above), namely that the triangle condition implies that χ(p) ≃ (pc−p) −1
as p ↑ pc; then we present a very brief discussion of the Hara–Slade verification
of the triangle condition for large d. We begin with a lemma.