PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

Percolation and Disordered Systems 265
Henceforth we assume that q ≥ 1. Turning to the question of phase transition,
and remembering percolation, we define the percolation probabilities
(13.8) θ b (p, q) = φ b p,q(0 ↔ ∞), b = 0, 1,
i.e., the probability that 0 belongs to an infinite open cluster. The corresponding
critical probabilities are given by
p b c (q) = sup{p : θb (p, q) = 0}, b = 0, 1.
Faced possibly with two (or more) distinct critical probabilities, we present
the following result, abstracted from [16, 158, 159, 162].
Theorem 13.9. Assume that d ≥ 2 and q ≥ 1. There exists a countable
subset P = Pq,d of [0, 1], possibly empty, such that φ 0 p,q = φ 1 p,q if either
θ 1 (p, q) = 0 or p /∈ P.
Consequently, θ0 (p, q) = θ1 (p, q) if p does not belong to the countable set
Pq,d, whence p0 c (q) = p1c (q). Henceforth we refer to the critical value as pc(q).
It is believed that Pq,d = ∅ for small q (depending on the value of d), and
that Pq,d = {pc(q)} for large q; see the next section.
Next we prove the non-triviality of pc(q) for q ≥ 1 (see [16]).
Theorem 13.10. If d ≥ 2 and q ≥ 1 then 0 < pc(q) < 1.
Proof. We compare the case of general q with the case q = 1 (percolation).
Using the comparison inequalities (Theorem 13.2), we find that
(13.11) pc(1) ≤ pc(q) ≤
qpc(1)
, q ≥ 1,
1 + (q − 1)pc(1)
where pc(1) is the critical probability of bond percolation on L d . Cf. Theorem
3.2. �
We note that pc(q) is monotone non-decreasing in q, by use of the comparison
inequalities. Actually it is strictly monotone and Lipschitz continuous
(see [161]).
Finally we return to the Potts model, and we review the correspondence
of phase transitions. The relevant ‘order parameter’ of the Potts model is
given by
M(βJ, q) = lim
Λ→Zd �
π 1 � � −1
Λ,β,J σ(0) = 1 − q �
,
where π1 Λ,β,J is a Potts measure on Λ ‘with boundary condition 1’. We may
think of M(βJ, q) as a measure of the degree to which the boundary condition
‘1’ is noticed at the origin. By an application of Theorem 12.8 to a suitable
graph obtained from Λ, we have that
π 1 � � −1 −1 1
Λ,β,J σ(0) = 1 − q = (1 − q )φΛ,p,q (0 ↔ ∂Λ)