PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

214 Geoffrey Grimmett
may be shown to be not too unreasonable, at least when d is not too large.
For further discussion, see [G].
We make some further points.
Universality. It is believed that the numerical values of critical exponents depend
only on the value of d, and are independent of the particular percolation
model.
Two dimensions. When d = 2, perhaps
α = −2 5
3 , β = 36
, γ = 43
18
91 , δ = 5 , . . .
Large dimension. When d is sufficiently large (actually, d ≥ dc) it is believed
that the critical exponents are the same as those for percolation on a tree
(the ‘mean-field model’), namely δ = 2, γ = 1, ν = 1 1
2 , ρ = 2 , and so on
(the other exponents are found to satisfy the scaling relations). Using the
first hyperscaling relation, this supports the contention that dc = 6. Such
statements are known to hold for d ≥ 19; see [179, 180] and Section 8.5.
8.4 Rigorous Results
Open challenges include to prove:
• the existence of critical exponents,
• universality,
• the scaling relations,
• the conjectured values when d = 2,
• the conjectured values when d ≥ 6.
Progress towards these goals has been slender, but positive. Most is known
in the case of large d, see the next section. For sufficiently large d, exact
values are known for many exponents, namely the values from percolation on
a regular tree. When d = 2, Kesten [205, 206] has proved that, if two critical
exponents exist, then certain others do also, and certain scaling relations are
valid. However, the provocative case when d = 3 is fairly open terrain.
Certain partial results are known in generality, yielding inequalities in situations
where one expects (asymptotic) equalities. For example, it is known
that β ≤ 1, if β exists (cf. (7.36)). In similar vein, we have that γ ≥ 1 and
δ ≥ 2 for all d.
8.5 Mean-Field Theory
The expression ‘mean-field’ permits several interpretations depending on context.
A narrow interpretation of the term ‘mean-field theory’ for percolation
involves trees rather than lattices. For percolation on a regular tree, it is
quite easy to perform exact calculations of many quantities, including the
numerical values of critical exponents. That is, δ = 2, γ = 1, ν = 1 1
2 , ρ = 2 ,
and other exponents are given according to the scaling relations (8.2); see
[G], Section 8.1.