PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

266 Geoffrey Grimmett
where p = 1 − e −βJ . Therefore
M(βJ, q) = (1 − q −1 ) lim
Λ→Zd φ 1 Λ,p,q (0 ↔ ∂Λ).
By an interchange of limits (which may be justified, see [16, 162]), we have
that11 lim
Λ→Zd φ 1 Λ,p,q (0 ↔ ∂Λ) = θ1 (p, q),
whence M(βJ, q) and θ 1 (p, q) differ only by the factor (1 − q −1 ).
13.3 First and Second Order Transitions
Let q ≥ 1 and 0 ≤ p ≤ 1. As before, φb p,q is the random-cluster measure
on Ld constructed according to the boundary condition b ∈ {0, 1}. The
corresponding percolation probability is θb (p, q) = φb p,q (0 ↔ ∞). There is a
phase transition at the point pc = pc(q). Much of the interest in Potts models
(and therefore random-cluster models) has been directed at a dichotomy in
the type of phase transition, which depends apparently on whether q is small
or large. The following picture is credible but proved only in part.
(a) Small q, say 1 ≤ q < Q(d). It is believed that φ 0 p,q = φ 1 p,q for all p, and
that θ b (pc(q), q) = 0 for b = 0, 1. This will imply (see [162]) that there is a
unique random-cluster measure, and that each θ b (·, q) is continuous at the
critical point. Such a transition is sometimes termed ‘second order’. The
two-point connectivity function
(13.12) τ b p,q (x, y) = φbp,q (x ↔ y)
satisfies
(13.13) − 1
n log τb p,q (0, ne1) → σ(p, q) as n → ∞
where σ(p, q) > 0 if and only if p < pc(q). In particular σ(pc(q), q) = 0.
(b) Large q, say q > Q(d). We have that φ0 p,q = φ1 p,q if and only if p �=
pc(q). When p = pc(q), then φ0 p,q and φ1p,q are the unique translationinvariant
random-cluster measures on Ld . Furthermore θ0 (pc(q), q) = 0 and
θ1 (pc(q), q) > 0, which implies that θ1 (·, q) is discontinuous at the critical
point. Such a transition is sometimes termed ‘first order’. The limit function
σ, given by (13.13) with b = 0, satisfies σ(pc(q), q) > 0, which is to say
that the measure φ0 p,q has exponentially decaying connectivities even at the
critical point. Trivially σ(p, q) = 0 when p > pc(q), and this discontinuity at
pc(q) is termed the ‘mass gap’.
11 We note that the corresponding limit for the free measure, limΛ→Z d φ 0 Λ,p,q (0 ↔
∂Λ) = θ 0 (p, q), has not been proved in its full generality; see [162, 315].