PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

212 Geoffrey Grimmett
8.1 Percolation Probability
8. CRITICAL PERCOLATION
The next main open question is to verify the following.
Conjecture 8.1. We have that θ(pc) = 0.
This is known to hold when d = 2 (using results of Harris [182], see
Theorem 9.1) and for sufficiently large values of d (by work of Hara and
Slade [179, 180]), currently for d ≥ 19. The methods of Hara and Slade
might prove feasible for values of d as small as 6 or 7, but not for smaller d.
Some new idea is needed for the general conclusion. As remarked in Section
7.4, we need to rule out the remaining theoretical possibility that there is an
infinite cluster in Z d when p = pc, but no infinite cluster in any half-space.
8.2 Critical Exponents
Macroscopic functions, such as the percolation probability, have a singularity
at p = pc, and it is believed that there is ‘power law behaviour’ at and near
this singularity. The nature of the singularity is supposed to be canonical,
in that it is expected to have certain general features in common with phase
transitions in other physical systems. These features are sometimes referred
to as ‘scaling theory’ and they relate to ‘critical exponents’.
There are two sets of critical exponents, arising firstly in the limit as
p → pc, and secondly in the limit over increasing distances when p = pc. We
summarise the notation in Table 8.1.
The asymptotic relation ≈ should be interpreted loosely (perhaps via logarithmic
asymptotics). The radius of C is defined by rad(C) = max{n : 0 ↔
∂B(n)}. The limit as p → pc should be interpreted in a manner appropriate
for the function in question (for example, as p ↓ pc for θ(p), but as p → pc
for κ(p)).
There are eight critical exponents listed in Table 8.1, denoted α, β, γ,
δ, ν, η, ρ, ∆, but there is no general proof of the existence of any of these
exponents.
8.3 Scaling Theory
In general, the eight critical exponents may be defined for phase transitions
in a quite large family of physical systems. However, it is not believed that
they are independent variables, but rather that they satisfy the following:
(8.2) Scaling relations
2 − α = γ + 2β = β(δ + 1),
∆ = δβ,
γ = ν(2 − η),