PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

164 Geoffrey Grimmett
s
1
θ = 0
pc(T) pc(L 2 )
θ > 0
1 p
Fig. 4.3. The critical ‘surface’. The area beneath the curve is the set of (p, s) for
which θ(p, s) = 0.
Theorem 4.7. It is the case that pc(T) < pc(L 2 ).
Sketch Proof of Theorem 4.7. Here is a rough argument, which needs some
rigour. There is a ‘critical curve’ in (p, s)-space, separating the regime where
θ(p, s) = 0 from that when θ(p, s) > 0 (see Figure 4.3). Suppose that this
critical curve may be written in the form h(p, s) = 0 for some increasing and
continuously differentiable function h. It is enough to prove that the graph
of h contains no vertical segment. Now
and, by Lemma 4.5,
whence
∇h =
∇h · (0, 1) = ∂h
∂s
1 ∂h
|∇h| ∂s =
��∂h ∂p
� �
∂h ∂h
,
∂p ∂s
≥ g(p, s)∂h
∂p ,
� � � 1
2
− 2
∂h
+ 1 ≥
∂s
g
� g 2 + 1 ,
which is bounded away from 0 on any closed subset of (0, 1) 2 . This indicates
as required that h has no vertical segment.
Here is the proper argument. There is more than one way of defining the
critical surface. Let Csub = {(p, s) : θ(p, s) = 0}, and let Ccrit be the set of
all points lying in the closure of both Csub and its complement.