PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

274 Geoffrey Grimmett
Turning to the square lattice, let Λn = [0, n] 2 , whose dual graph Λd n may
1
2 , 2 ) by identifying all boundary vertices. This
be obtained from [−1, n] 2 + ( 1
implies by (13.32) that
(13.36) φ 0 Λn,p,q (ω) = φ1
Λd n ,pd ,q (ωd )
for configurations ω on Λn (and with a small ‘fix’ on the boundary of Λ d n ).
Letting n → ∞, we obtain that
(13.37) φ 0 p,q(A) = φ 1
p d ,q (Ad )
for all cylinder events A, where A d = {ω d : ω ∈ A}.
As a consequence of this duality, we may obtain as in the proof of Theorem
9.1 that
(13.38) θ 0� κq, q � = 0
(see [162, 345]), whence the critical value of the square lattice satisfies
(13.39) pc(q) ≥
It is widely believed that
pc(q) =
√ q
1 + √ q
√ q
1 + √ q
for q ≥ 1.
for q ≥ 1.
This is known to hold when q = 1 (percolation), when q = 2 (Ising model),
and for sufficiently large values of q. Following the route of the proof of
Theorem 9.1, it suffices to show that
φ 0 � � −nψ(p,q)
p,q 0 ↔ ∂B(n) ≤ e
for all n,
and for some ψ(p, q) satisfying ψ(p, q) > 0 when p < pc(q). (Actually, rather
less than exponential decay is required; it would be enough to have decay at
rate n−1 .) This was proved by the work of [13, 236, 331] when q = 2. When q
is large, this and more is known. Let µ be the connective constant of L2 , and
let Q = � � � ��
1
2 µ + µ 2 4.
− 4 We have that 2.620 < µ < 2.696 (see [334]),
whence 21.61 < Q < 25.72. We set
ψ(q) = 1
24 log
noting that ψ(q) > 0 if and only if q > Q.
� (1 + √ q) 4
qµ 4
�
,
Theorem 13.40. If d = 2 and q > Q then the following hold.
(a) The critical point is given by pc(q) = √ q/(1 + √ q).