CRITERIO DE ACEPTACIÓN
4.2.2. Matriz de Trazabilidad
step must be performed properly (see [55, 106]), but the following rough argument is appealing and may be made rigorous. Select a trifurcation (t1, say) of S, and choose some vertex y1∈∂S such that t1↔y1in S. We now select a new trifurcation t2 ∈ S. It may be seen, using the definition of the term ‘trifurcation’, that there exists y2∈ ∂S such that y16=y2and t2↔ y2in S. We continue similarly, at each stage picking a new trifurcation tk ∈S and a new vertex yk ∈∂S. If there areτ trifurcations in S, then we obtainτ distinct vertices yk of∂S. Therefore,|∂S| ≥τ. However, by the
remarks above,Ep(τ )is comparable to S. This is a contradiction for large
n, since|∂S|grows in the manner of nd−1and|S|grows in the manner of
nd.
5.4 Phase transition
Macroscopic functions, such as the percolation probability and mean cluster size,
θ (p)=Pp(|C| = ∞), χ (p)=Ep|C|,
have singularities at p = pc, and there is overwhelming evidence that these are of ‘power law’ type. A great deal of effort has been invested by physicists and mathematicians towards understanding the nature of the percolation phase-transition. The picture is now fairly clear when d =2, owing to the very significant progress in recent years in relating critical percolation to the Schramm–L ¨owner curve SLE6. There remain however substantial difficulties to be overcome before this chapter of percolation theory can be declared written, even when d = 2. The case of large d (currently, d ≥ 19) is also well understood, through work based on the so-called ‘lace expansion’. Most problems remain open in the obvious case
d=3, and ambitious and brave students are thus directed with caution. The nature of the percolation singularity is supposed to be canonical, in that it is expected to have certain general features in common with phase transitions of other models of statistical mechanics. These features are sometimes referred to as ‘scaling theory’ and they relate to ‘critical expo- nents’. 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 summarize the notation in Table 5.7.
The asymptotic relation ≈ should be interpreted loosely (perhaps via logarithmic asymptotics1). The radius of C is defined by
rad(C)=sup{kxk: 0↔x},
1We say that f(x)is logarithmically asymptotic to g(x)as x → 0 (respectively,
Function Behaviour Exp.
percolation
probability θ (p)=Pp(|C| = ∞) θ (p)≈(p−pc)β β
truncated
mean cluster size χf(p)=Ep(|C|1|C|<∞) χf(p)≈ |p−pc|−γ γ
number of
clusters per vertex κ(p)=Ep(|C|−1) κ′′′(p)≈ |p−pc|−1−α α
cluster moments χkf(p)=Ep(|C|k1|C|<∞) χkf+1(p) χkf(p) ≈ |p−pc| −1,k≥1 1 correlation length ξ(p) ξ(p)≈ |p−pc|−ν ν cluster volume Pp c(|C| =n)≈n−1−1/δ δ cluster radius Pp c rad(C)=n ≈n−1−1/ρ ρ connectivity function Pp c(0↔x)≈ kxk2−d−η η
Table 5.7. Eight functions and their critical exponents.
where
kxk =sup
i |
xi|, x=(x1,x2, . . . ,xd)∈Zd,
is the supremum (L∞) norm on Zd. The limit as p → pc should be interpreted in a manner appropriate for the function in question (for example, as p↓pcforθ (p), but as p→ pcforκ(p)).
There are eight critical exponents listed in Table 5.7, denotedα,β,γ, δ,ν,η,ρ,1, but there is no general proof of the existence of any of these exponents for arbitrary d. 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
5.4 Phase transition 97 that they satisfy the scaling relations
2−α=γ +2β =β(δ+1), 1=δβ,
γ =ν(2−η),
and, when d is not too large, the hyperscaling relations
dρ=δ+1, 2−α=dν.
The upper critical dimension is the largest value dcsuch that the hyperscaling relations hold for d ≤dc. It is believed that dc=6 for percolation. There is no general proof of the validity of the scaling and hyperscaling relations, although quite a lot is known when d=2 and for large d.
In the context of percolation, there is an analytical rationale behind the scaling relations, namely the ‘scaling hypotheses’ that
Pp(|C| =n)∼n−σf n/ξ(p)τ Pp(0↔x,|C|<∞)∼ kxk2−d−ηg kxk/ξ(p)
in the double limit as p → pc, n → ∞, and for some constantsσ,τ,η and functions f , g. Playing loose with rigorous mathematics, the scaling relations may be derived from these hypotheses. Similarly, the hyperscaling relations may be shown to be not too unreasonable, at least when d is not too large. For further discussion, see [106].
We note 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 perco- lation model.
Two dimensions. When d=2, perhaps
α= −23, β= 365, γ =4318, δ=915, . . . See (5.45).
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’), namelyδ = 2,γ = 1, ν = 12,ρ = 12, and so on (the other exponents are found to satisfy the scaling relations). Using the first hyperscaling relation, this is consistent with the contention that dc=6. Such statements are known to hold for d≥19, see [132, 133] and the remarks later in this section.
Open challenges include to prove: – the existence of critical exponents, – universality,
– the scaling and hyperscaling relations, – the conjectured values when d=2, – the conjectured values when d≥6.
Progress towards these goals has been positive. For sufficiently large d, exact values are known for many exponents, namely the values from per- colation on a regular tree. There has been remarkable progre ss in recent years when d = 2, inspired largely by work of Schramm [215], enacted by Smirnov [222], and confirmed by the programme pursued by Lawler, Schramm, and Werner to understand SLE curves. See Section 5.6.
We close this section with some further remarks on the case of large d. The expression ‘mean-field’ permits several interpretations depending on context. A narrow interpretation of the term ‘mean-field theory’ for perco- lation 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 = 12, ρ=12, and other exponents are given according to the scaling relations, see [106, Chap. 10].
Turning to percolation on Ld, it is known as remarked above that the critical exponents agree with those of a regular tree when d is sufficiently large. In fact, this is believed to hold if and only if d ≥6, but progress so far assumes that d≥19. In the following theorem, we write f(x)≃g(x) if there exist positive constants c1,c2such that c1f(x)≤ g(x)≤c2f(x) for all x close to a limiting value.
5.26 Theorem [133]. For d≥19,