PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

272 Geoffrey Grimmett
which, via Lemma 13.22, implies as in (13.25) that
φs,q(Hi) ≥
D(log n)2
(log log n) ∆4
for some D > 0. By (13.28), and the fact that K = ⌊n/m⌋,
φs,q(Fn) ≥
D ′ n
(log log n) ∆4
for some D ′ > 0. By (13.24),
�
c5n
φr,q(An) ≤ exp −
(loglog n) ∆4
Since ∆4 > 1, this implies the claim of the theorem with k = 1. The claim
for general k requires k − 1 further iterations of the argument.
(b) We omit the proof of this part. The fundamental argument is taken from
[138], and the details are presented in [167]. �
13.5 The Case of Two Dimensions
In this section we consider the case of random-cluster measures on the square
lattice L 2 . Such measures have a property of self-duality which generalises
that of bond percolation. We begin by describing this duality.
Let G = (V, E) be a plane graph with planar dual G d = (V d , E d ). Any
configuration ω ∈ ΩE gives rise to a dual configuration ω d ∈ Ω E d defined as
follows. If e (∈ E) is crossed by the dual edge e d (∈ E d ), we define ω d (e d ) =
1−ω(e). As usual, η(ω) denotes the set {e : ω(e) = 1} of edges which are open
in ω. By drawing a picture, one may be convinced that every face of (V, η(ω))
contains a unique component of (V d , η(ω d )), and therefore the number f(ω)
of faces (including the infinite face) of (V, η(ω)) satisfies f(ω) = k(ω d ). See
Figure 13.1. (Note that this definition of the dual configuration differs slightly
from that used earlier for two-dimensional percolation.)
The random-cluster measure on G is given by
Using Euler’s formula,
φG,p,q(ω) ∝
�
.
� � |η(ω)|
p
q
1 − p
k(ω) .
k(ω) = |V | − |η(ω)| + f(ω) − 1,
and the facts that f(ω) = k(ω d ) and |η(ω)| + |η(ω d )| = |E|, we have that
φG,p,q(ω) ∝
� q(1 − p)
p
� |η(ω d )|
q k(ωd ) ,