PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

176 Geoffrey Grimmett
6.1 Using Subadditivity
6. SUBCRITICAL PERCOLATION
We assume throughout this chapter that p < pc. All open clusters are a.s.
finite, and the phase is sometimes called ‘disordered’ by mathematical physicists,
since there are no long-range connections. In understanding the phase,
we need to know how fast the tails of certain distributions go to zero, and
a rule of thumb is that ‘everything reasonable’ should have exponentially
decaying tails. In particular, the limits
�
φ(p) = lim −
n→∞
1 � �
log Pp 0 ↔ ∂B(n)
n �
,
�
ζ(p) = lim −
n→∞
1
n log Pp(|C|
�
= n) ,
should exist, and be strictly positive when p < pc. The function φ(p) measures
a ‘distance effect’ and ζ(p) a ‘volume effect’.
The existence of such limits is a quite different matter from their positiveness.
Existence is usually proved by an appeal to subadditivity (see below)
via a correlation inequality. To show positiveness usually requires a hard
estimate.
Theorem 6.1 (Subadditive Inequality). If (xr : r ≥ 1) is a sequence of
reals satisfying the subadditive inequality
then the limit
xm+n ≤ xm + xn for all m, n,
�
1
λ = lim
r→∞ r xr
�
exists, with −∞ ≤ λ < ∞, and satisfies
�
1
λ = inf
r xr
�
: r ≥ 1 .
The history here is that the existence of exponents such as φ(p) and
ζ(p) was shown using the subadditive inequality, and their positiveness was
obtained under extra hypotheses. These extra hypotheses were then shown
to be implied by the assumption p < pc, in important papers of Aizenman
and Barsky [12] and Menshikov [267, 270]. The case d = 2 had been dealt
with earlier by Kesten [201, 203].
As an example of the subadditive inequality in action, we present a proof
of the existence of φ(p) (and other things . . . ). The required ‘hard estimate’
is given in the next section. We denote by e1 a unit vector in the direction
of increasing first coordinate.