PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

228 Geoffrey Grimmett
x
Fig. 9.3. If there is an open left-right crossing of the box, then there must exist
some vertex x in the centre which is connected disjointly to the left and right sides.
circuit surrounding the origin and contained within B(n). In particular, no
dual annulus of the form B(3r+1 )\B(3r ) + ( 1 1
2 , 2 ), for 0 ≤ r < log3 n − 1, can
contain such a closed circuit. Therefore
� � log
(9.7) P1 0 ↔ ∂B(n) ≤ (1 − ξ) 3 n−2
2
as required for the upper bound in (9.6). �
9.3 Conformal Invariance
We concentrate on bond percolation in two dimensions with p = pc = 1
2 .
With S(n) = [0, n + 1] × [0, n], we have by self-duality that
� �
1
(9.8) P1 S(n) traversed from left to right by open path =
2
2
for all n. Certainly L2 must contain long open paths, but no infinite paths
(since θ( 1
2 ) = 0). One of the features of (hypothetical) universality is that
the chances of long-range connections (when p = pc) should be independent
of the choice of lattice structure. In particular, local deformations of
space should, within limits, not affect such probabilities. One family of local
changes arises by local rotations and dilations, and particularly by applying
a conformally invariant mapping to R2 . This suggests the possibility that
long-range crossing probabilities are, in some sense to be explored, invariant
under conformal maps of R2 . (See [8] for an account of conformal maps.)
Such a hypothesis may be formulated, and investigated numerically. Such
a programme has been followed by Langlands, Pouliot, and Saint-Aubin [232]
and Aizenman [10], and their results support the hypothesis. In this summary,
we refer to bond percolation on L2 only, although such conjectures
may be formulated for any two-dimensional percolation model.