Tabla periódica de los elementos

In the next chapters, we investigate several different geometric hyperedge potentials for background and type interactions. For each of these models, we will establish the existence of percolation in the geometric continuum random cluster modelCΛ|ξ when q ≥ 1 is an

arbitrary parameter,zis sufficiently large and for other appropriately chosen parameters, for the pseudo-periodic boundary conditionξ∈Γb_Λc whereΓbwas defined in (2.19) andM

andΓare chosen such that(U)is satisfied. More precisely, let

Λn:=

[

k∈{−n,...,n}2 ∇(k).

for largen ∈ N, where∇(k)is given in (2.18). We show that for any∆∈ B(R2), there existsc >0, such that for allΛn⊃∆, and for allξ = (ξ, σξ)∈bΓ_Λc

Z

CΛn|ξ(dω, dE)N∆↔Λcn(ω, E)≥c. (2.40)

This is the main result upon which the proof of the phase transition in all our different mod- els is based on, and is analogous to the statement of Proposition 3.1 in [GH96], but adapted to the hypergraph structure. The proof of (2.40) is deferred until later, but supposing it is satisfied, we can relate it to the notion of a phase transition in the geometric continuum Potts model. We give an outline of the main arguments here.

Assuming that the couple(Φ,H)satisfy(R),(S)and(U), a Gibbs measure is con- structed as a limit of Gibbs distributions in boxesΛn. Letξ = (ξ, σξ) ∈ bΓ_Λc

n such that

σξ(x) = 1 for allx ∈ ξ be a pseudo-periodic boundary condition with mark 1. The up-

per regularity(U)and stability(S)conditions then show thatξis admissible forΛnandz.

Therefore,

Qn:=Q_Λn|ξ◦pr−_Λn1

call the matrixM ∈R2×2from (2.18) and letPnbe the probability measure on(Ω,F)such

that the marked configurations in the disjoint boxesΛn+(2n+1)M k, k∈Z2, are indepen-

dent with respect toPn and have distributionQn. Rather than obtaining a full asymptotic

translation invariance now, we first confine ourselves to the skewed lattice translations, in recognition of our cell structure. So we define the spatial average ofPn,

b Pn= 1 |Λn| X i∈L(n) Pn◦ϑ−i 1,

whereL(n) = Λn∩MZ2. To obtain a limit for the sequence(Pb_n)n≥1, a suitable topology

must be specified. A measurable functionf : Ω→Ris called local and tame if

f(ω) =f(ωΛ) and |f(ω)| ≤aNΛ(ω) +b

for allω = (ω, σω)∈Ω, someΛ∈ B(R2)and suitable constantsa, b≥0, whereNΛ(ω)is

the counting variable defined in (2.1). The set of all local and tame functions is denoted by

T. TheT–topology, on the set of all translation invariant probability measures on(Ω,F), is defined as the smallest topology for which the mappingsP 7→ R

f dP are continuous, wheref ∈ T. It can be shown that there exists a subsequence of(Pb_n)n≥1which converges

in theT–topology. The limit of this subsequence Pb, once found, cannot be shown to be

concentrated on admissible configurations, see [DDG10], and so it is not known to be Gibbs. However, sincePbis non-degenerate, it follows by the Range condition(R)and Remark 2.9

that

P :=Pb(·|{∅}c)

is a Gibbs measure and hence, a Delaunay continuum Potts measure forH,Φandz, and invariant under the skewed lattice translations. To obtain full translation invariance under

(ϑx)x∈R2, the spatial average ofP,

P(1) := 1

|∇(0)| Z

∇(0)

P◦ϑ−_x1dx,

is taken. The superscript identifies the choice for the marks of the particles in the boundary when constructingPb_n. An application of (2.40) and Proposition 2.37 shows that

Z (qN∆,1−N∆)dPb_n= (q−1) Z N∆↔ΛcdC_Λ n|ξ (2.41) ≥(q−1)c. (2.42)

P(1)is symmetric under changes to the marks2, . . . , q, this means that Z (qN∆,1−N∆)dP(1) > Z (qN∆,2−N∆)dP(1) =· · ·= Z (qN∆,q−N∆)dP(1).

Fort∈ {2, . . . , q}, letP(t)be the translation invariant, Delaunay continuum Potts measure which is obtained fromP(1)_{by switching the marks1andt. It is then evident, that there ex-}

istsqdistinct translation invariant, Delaunay continuum Potts measures – a phase transition.

The only missing step so far, is the proof of (2.40), i.e. the existence of percolation in the Delaunay continuum random cluster models. This depends heavily on the specific model in question, and is the dedicated work of this thesis – it forms the bulk of Chapters 3 and 4. Although we use different techniques for different classes of model, we present here a theme that runs through each: a relationship involving the existence of percolation for geometric continuum random cluster distributionsCΛ|ξand the existence of percolation

in certain site percolation models. LetM_Λ|ξ be the distribution of particle positions given

by the marginal distributionCΛ|ξ(·,E). We can then writeCΛ|ξas follows:

CΛ|ξ(dω, dE) =MΛ|ξ(dω)µ(_ω,q)_Λ(dE), (2.43) with µ(_ω,q)_Λ(dE) = q K(ω,E)_µ ω,Λ(dE) R qK(ω,E)_µ ω,Λ(dE) . (2.44)

Letµ˜ω be an alternative distribution of open hyperedge configurations where each hyper-

edgeη∈ H(ω)is declared open with probability

pΛ(η) =p1H∗_(ω)(η), (2.45)

and closed otherwise, where H∗_{(ω) is some subset of H(ω) andp ∈ [0,1]. Instead of}

declaring hyperedges as open or closed, we may wish, instead, to declare the mark of each particle. This leads us to the definition of a continuum site percolation model,C˜_Λsite_|ξ, with the same distribution of particle positions as our geometric continuum random cluster model. We write

˜

C_Λsite_|ξ(dω) =M_Λ|ξ(dω)˜λω(dω), (2.46)

elements inΣ, where(σω(x))x∈ωare independent random variables with probability

P rob(σω(x) = 1) =p1ω∗(x). (2.47)

Using the same arguments as in Lemma 2.13, we see that the map ω → λ˜ω is a

probability kernel fromΩtoΩ. Note that the value ofpis the same as in (2.45) andω∗ ⊂ω

is the set of points ofωthat build the hyperedges in H∗(ω). Finally, define the event that there is a path that intersects both∆andΛcand is made up of points of mark1connected by edges in the reduced hypergraphH∗(ω). That is:

{∆↔Λc}:= n ω∈Ω× {1}:∃x1, . . . , xn∈ωwithx1∈∆, xn∈Λc and fori= 1, . . . , n−1, ∃η ∈ H∗(ω) :{xi, xi+1} ⊆η o . (2.48)

Proposition 2.18. Letµ˜ω and˜λω be as in (2.45) and (2.47). Then, ifµ

(q)

ω,Λ <µ˜ω, we have

for all∆,Λ∈ B(R2), with∆⊂Λ

Z N∆↔ΛcdC_Λ|ξ ≥C˜site Λ|ξ({∆↔Λc}). Proof. Z C_Λ|ξ(dω, dE)N∆↔Λc(ω, E) = Z M_Λ|ξ(dω) Z µ(_ω,q)_Λ(dE)N∆↔Λc(ω, E) ≥ Z MΛ|ξ(dω) Z ˜ µω(dE)N∆↔Λc(ω, E) ≥ Z MΛ|ξ(dω) Z ˜ µω(dE)1{N∆↔Λc≥1}(ω, E) ≥ Z M_Λ|ξ(dω) Z ˜ λω(dσω)1{∆↔Λc_}(ω, σ_ω) = ˜C_Λsite_|ξ({∆↔Λc}),

where the first inequality is a direct application of the assumption in the statement of the Proposition (sinceN∆↔Λc is an increasing event), and the third inequality is due to the fact that site percolation implies hyperedge percolation.

We have shown that to prove the existence of percolation in a geometric continuum random cluster model, it is sufficient to show percolation in a suitable continuum site model. This is essentially the work of the next two chapters as we investigate several different geometric hyperedge potentials for background and type interactions.