PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

162 Geoffrey Grimmett
Fig. 4.1. An entanglement between opposite sides of a cube in three dimensions.
Note the necklace of necklaces on the right.
be seen that there is an ‘entanglement transition’ at some critical point pe
satisfying pe ≤ pc. Is it the case that strict inequality holds, i.e., pe < pc?
A technology has been developed for approaching such questions of strict
inequality. Although, in particular cases, ad hoc arguments can be successful,
there appears to be only one general approach. We illustrate this approach
in the next section, by sketching the details in a particular case.
Important references include [19, 156, 157, 268]. See also [74].
4.3 The Square and Triangular Lattices
The triangular lattice T may be obtained by adding diagonals across the
squares of the square lattice L 2 , in the manner of Figure 4.2. Since any
infinite open cluster of L 2 is also an infinite open cluster of T, it follows that
pc(T) ≤ pc(L 2 ), but does strict inequality hold? There are various ways of
proving the strict inequality. Here we adopt the canonical argument of [19],
as an illustration of a general technique.
Before embarking on this exercise, we point out that, for this particular
case, there is a variety of ways of obtaining the result, by using special
properties of the square and triangular lattices. The attraction of the method
described here is its generality, relying as it does on essentially no assumptions
about graph-structure or number of dimensions.
First we embed the problem in a two-parameter system. Let 0 ≤ p, s ≤ 1.
We declare each edge of L 2 to be open with probability p, and each further
edge of T (i.e., the dashed edges in Figure 4.2) to be open with probability
s. Writing Pp,s for the associated measure, define
θ(p, s) = Pp,s(0 ↔ ∞).