PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

or
Percolation and Disordered Systems 211
θ ′ (π) + θ(π) ≥ 1.
Integrate this over the interval (pc, p) to obtain
θ(p)e p − θ(pc)e pc ≥ e p − e pc , pc ≤ p,
whence it is an easy exercise to show that
(7.36) θ(p) − θ(pc) ≥ a(p − pc), pc ≤ p,
for some positive constant a. The above argument may be made rigorous.
Differential inequalities of the type above are used widely in percolation
and disordered systems.
7.6 Cluster-Size Distribution
When p < pc, the tail of the cluster-size |C| decays exponentially. Exponential
decay is not correct when p > pc, but rather ‘stretched exponential
decay’.
Theorem 7.37. Suppose pc < p < 1. There exist positive constants α(p),
β(p) such that, for all n,
(7.38) exp � −αn (d−1)/d� ≤ Pp(|C| = n) ≤ exp � −βn (d−1)/d� .
See [G] for a proof of this theorem. The reason for the power n (d−1)/d
is roughly as follows. It is thought that a large finite cluster is most likely
created as a cluster of compact shape, all of whose boundary edges are closed.
Now, if a ball has volume n, then its surface area has order n (d−1)/d . The
price paid for having a surface all of whose edges are closed is (1 −p) m where
m is the number of such edges. By the above remark, m should have order
n (d−1)/d , as required for (7.38).
It is believed that the limit
�
1
(7.39) γ(p) = lim −
n→∞ n (d−1)/d log Pp(|C|
�
= n)
exists, but no proof is known.
Much more is known in two dimensions than for general d. The size and
geometry of large finite clusters have been studied in detail in [37], where it
was shown that such clusters may be approximated by the so called ‘Wulff
shape’. This work includes a proof of the existence of the limit in (7.39) when
d = 2.