PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

Percolation and Disordered Systems 229
We begin with a concrete conjecture concerning crossing probabilities. Let
B(kl, l) be a 2kl by 2l rectangle, and let LR(kl, l) be the event that B(kl, l)
is traversed between its opposite sides of length 2l by an open path, as in
Section 9.2. It is not difficult to show, using (9.8), that
� �
1
P1 LR(l, l) →
2
2
and it is reasonable to conjecture that the limit
as l → ∞,
� �
(9.9) λk = lim P1 LR(kl, l)
l→∞ 2
exists for all 0 < k < ∞. By self-duality, we have that λk + λ k −1 = 1 if the
λk exist. It is apparently difficult to establish the limit in (9.9).
In [232] we see a generalisation of this conjecture which is fundamental
for a Monte Carlo approach to conformal invariance. Take a simple closed
curve C in the plane, and arcs α1, α2, . . . , αm, β1, . . . , βm, as well as arcs
γ1, γ2, . . . , γn, δ1, . . . , δn, of C. For a dilation factor r, define
(9.10) πr(G) = P(rαi ↔ rβi, rγi � rδi, for all i, in rC)
where P = Ppc and G denotes the collection (C; αi, βi; γi, δi).
Conjecture 9.11. The following limit exists:
π(G) = lim
r→∞ πr(G).
Some convention is needed in order to make sense of (9.10), arising from
the fact that rC lives in the plane R 2 rather than on the lattice L 2 ; this
poses no major problem. Conjecture (9.9) is a special case of (9.11), with
C = B(k, 1), and α1, β1 being the left and right sides of the box.
Let φ : R 2 → R 2 be a reasonably smooth function. The composite object
G = (C; αi, βi; γi, δi) has an image under φ, namely the object φG =
(φC; φαi, φβi; φγi, φδi), which itself corresponds to an event concerning the
existence or non-existence of certain open paths. If we believe that crossing
probabilities are not affected (as r → ∞, in (9.10)) by local dilations and
rotations, then it becomes natural to formulate a conjecture of invariance
under conformal maps [10, 232].
Conjecture 9.12 (Conformal Invariance). For all G = (C; αi, βi; γi, δi),
we have that π(φG) = π(G) for any φ : R 2 → R 2 which is bijective on C and
conformal on its interior.
Lengthy computer simulations, reported in [232], support this conjecture.
Particularly stimulating evidence is provided by a formula known as Cardy’s
formula [86]. By following a sequence of transformations of models, and
applying ideas of conformal field theory, Cardy was led to an explicit formula