PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

158 Geoffrey Grimmett
Kesten [201] proved that pc(L2 ) = 1
2 . This very special calculation makes
essential use of the self-duality of L2 (see Chapter 9). There are various ways
of proving the strict inequality
pc(L d ) − pc(L d+1 ) > 0 for d ≥ 2,
and good recent references include [19, 157].
On the third point above, we point out that
pc(L d ) = 1
�
1 7 1 1
+ + + O
2d (2d) 2 2 (2d) 3 (2d) 4
�
as d → ∞.
See [179, 180, 181], and earlier work of [150, 210].
We note finally the canonical arguments used to establish the inequality
0 < pc(L d ) < 1. The first inequality was proved by counting paths, and the
second by counting circuits in the dual. These approaches are fundamental
to proofs of the existence of phase transition in a multitude of settings.
3.3 A Question
The definition of pc entails that
�
= 0 if p < pc,
θ(p)
> 0 if p > pc,
but what happens when p = pc?
Conjecture 3.8. θ(pc) = 0.
This conjecture is known to be valid when d = 2 (using duality, see Section
9.1) and for sufficiently large d, currently d ≥ 19 (using the ‘bubble
expansion’, see Section 8.5). Concentrate your mind on the case d = 3.
Let us turn to the existence of an infinite open cluster, and set
ψ(p) = Pp(|C(x)| = ∞ for some x).
By using the usual zero-one law (see [169], p. 290), for any p either ψ(p) = 0
or ψ(p) = 1. Using the fact that Z d is countable, we have that
ψ(p) = 1 if and only if θ(p) > 0.
The above conjecture may therefore be written equivalently as ψ(pc) = 0.
There has been progress towards this conjecture: see [49, 164]. It is
proved that, when p = pc, no half-space of Z d (where d ≥ 3) can contain
an infinite open cluster. Therefore we are asked to eliminate the following
absurd possibility: there exists a.s. an infinite open cluster in L d , but any
such cluster is a.s. cut into finite parts by the removal of all edges of the form
〈x, x + e〉, as x ranges over a hyperplane of L d and where e is a unit vector
perpendicular to this hyperplane.