Two Pure Infinite Clusters

This subsection analyses a Widom-Rowlinson measure with two pure infinite clus- ters.

Let us begin with the absurd case.

Lemma 4.37 Let λ >0. The eventE−∗∩F0_∩E+∗

is impossibleWR∗(λ)-almost- surely, i.e., for any Widom Rowlinson measures µ, the event E−∗∩F0_∩E+∗ _has

Proof: Assume for contradiction that there exists an extremal Widom-Rowlinson measure µ∈ WR∗_EX(λ) with µ(E−∗∩F0 _∩E+∗_{) >0}

and fix it. By tail triviality of µ, the tail-event F0 _{occurs µ-almost surely, which implies that any finite sub-}

set ∆_b Z2 is encircled by a −+∗circuit, i.e., a ∗circuit equipped with −spins or

+spins. Since two ∗adjacent nodes never have strict opposite spin valuesµ-almost surely, we can even state that any finite subset ∆_bZ2 _{is encircled by a−∗circuit}

or a +∗circuit µ-almost surely. But this is a contradiction either to the existence of an infinite −∗cluster or to the existence of an infinite +∗cluster depending on

which ∗circuit occurs infinitely often.

Since the remaining two cases are similar, we deal with them in one proposition. Proposition 4.38 Let λ >0. If

lim inf

Λ%_Z2

∂∗_{Λis a circuit}

µ0∗_∆,λ(∃ ± ∗lasso in∆) >0,

then the setE−∗∩E0_∩F+∗_∪F−∗_∩E0_∩E+∗ _{is impossible WR}∗

(λ)-almost surely. Proof: We restrict ourselves to show that the set F−∗ ∩E0 _∩E+∗

is impossible WR∗(λ)-almost surely, since the rest of the statement follows by symmetry. By extremal decomposition, it is sufficient to show that for any extremal Widom- Rowlinson measure µ ∈ WR∗_EX(λ), the event F−∗ ∩E0 _∩E+∗

has µ-probability zero.

Let us begin by verifying that F−∗ ∩E0∩E+∗ is a tail-event WR∗(λ)-almost surely, which will be done by applying the Shield Lemma of Chapter 3, see page 20. To this end, we map the configurations of{−1,0,1}Z2 to {0,1}Z2 by flipping all−spins and denote this map by m−7→+. Since all finite −∗clusters are encircled

by 0circuits WR∗(λ)-almost surely, the event m−7→+(F−∗ ∩E0∩E+∗) exhibits a

sole infinite 0cluster and a sole infinite 1∗cluster. Therefore, all conditions – the uniqueness of both infinite clusters – of the Shield Lemma are met and, given the “flipped“ eventm−7→+(F−∗∩E0∩E+∗), any finite subset ∆bZ2 is encircled by a

mixed 1∗₀ circuit. Consequently, given the event F−∗ ∩E0_∩E+∗_{, any finite subset}

∆_bZ2is encircled by a mixed+∗₀ circuit WR∗(λ)-almost surely, since any sequence of subsets tending toZ2 is met by the infinite +∗cluster eventually. Summing up, we can characterise the eventF−∗∩E0_∩E+∗ _{by the intersection of the tail-events}

”an infinite 0cluster occurs“, ”an infinite +∗cluster occurs“ and ”any finite subset

∆_bZ2 _{is encircled by a mixed}+∗

0 circuit“, which implies thatF

−∗_∩E0_∩E+∗

itself is a tail-event WR∗(λ)-almost surely.

Hence, fix a measureµ∈WR∗_EX(λ)and let us assume for contradictionµ(F−∗∩

E0_∩E+∗_{) = 1}

. Recall that there exists an >0 so that

lim inf

Λ%_Z2

∂∗Λis a circuit

Fix an arbitrary node x∈Z2_{. Inequality (4.16), together with symmetry, enables}

us to fix a Γ_bZ2 _{with~0, x∈Γso that for any ∆with ∂}∗_∆

is a circuit encircling

Γ

µ0∗_∆,λ(∃ +∗lasso around x in∆) ≥/3. (4.17) Since all −∗clusters are finite µ-almost surely, for any ∆ _b Z2 _{with ∂}∗_{∆ is a}

circuit encircling Γ, there exists a∆0 _bZ2 larger than ∆so that with probability at least 1/2, one can find a 0+circuit in ∆0 around ∆. Considering the strong Markov property entails the following identities

0< c:=µ+∗_λ (0←→ ∞+∗ )/6 ≤µ+∗_λ (x←→ ∞+∗ )(/3)µ(C_∆max 0+0_\∆ 6=∅) =µ+∗_λ (x←→ ∞+∗ ) Z µ(dω)_1{Cmax 0+ ∆0\∆ 6=∅}(ω)(/3) (4.17) ≤ µ+∗_λ (x←→ ∞+∗ ) Z µ(dω)_1{Cmax 0+ ∆0\∆ 6=∅} (ω) µω_int∗_Cmax 0+ ∆0\∆ ,λ (∃+∗lasso aroundx) = Z µ(dω)_1{Cmax 0+

∆0\∆ 6=∅,∃+∗lasso aroundxin intC max 0+ ∆0\∆ }(ω)µ +∗ λ (x +∗ ←→ ∞) ≤ Z µ(dω)_1{Cmax 0+

∆0\∆ 6=∅,∃+∗lasso aroundxin intC max 0+ ∆0\∆ } (ω) µ+∗_λ (x←→+∗ Cmax +∗ intCmax 0+ ∆0\∆ (ω) (ω)) ≤ Z µ(dω)_1{Cmax 0+

∆0\∆ 6=∅,∃+∗lasso aroundxin intC max 0+ ∆0\∆ } (ω) µω_int∗_Cmax +∗ intCmax 0+ ∆0\∆ (x←→+∗ Cmax +∗ intC_∆max 0+_0\∆ )

=µ(C_∆max 0+0_\∆ 6=∅,∃+∗lasso around x in intC_∆max 0+0_\∆ , x

+∗ ←→Cmax +∗ intCmax 0+ ∆0\∆ ) ≤µ(x←→+∗ ∆c),

where the last but one inequality follows from Lemma (4.32) and positive associ- ation of µ+∗_λ . Summing up, if ∆ tends to Z2_{, then with probability at leastc > 0,}

any x is contained in the infinite +∗cluster, i.e., for any x∈Z2

µ(x←→ ∞+∗ )≥c .

To apply Theorem 3.3 on page 19 we have to map {−1,0,1}Z2 to {0,1}Z2

m−7→0. The advantage of this kind of mapping is that the event {m−7→0 ∈ A}

is increasing if A is increasing and, therefore, the measure µ0 := µ◦m−1_−7→0 has positive associations. Furthermore, there still exists one infinite 1∗cluster as well as one infinite0clusterµ0-almost surely, since every finite−∗cluster is encircled by a 0circuit µ-almost surely. By construction of µ0, the bounded energy property is also satisfied. This is the case because of the following two facts:

a) every0spin[m−7→0(ξ)]of a mapped configurationm−7→0(ξ)could be a0spin[ξ]

in the underlying configuration ξ;

b) for any node y, the µ-probability that y takes spin value 0 or + is strictly positive if we condition on the event that ∂∗y is equipped with 0or +spins. Thus, we can apply Theorem 3.3 and obtain a contradiction to our assumption

µ(F−∗∩E0∩E+∗) = 1.