Las destrezas comunicativas.

This section presents a new sufficient condition for the absence of phase transition, closely related to µ+∗_λ (x ←→ ∞+∗ ). At the end the analogous result regarding the absence of phase transition on the graph (Z2_,

) will be stated.

First, let us recall the following notation. In order to specify the underlying configuration of a mathematical object we add the configuration in brackets, e.g., we write “the 0circuit[σ]” instead of “the 0circuit w.r.t.σ”.

To prepare the proof of the main result of this section, we first split the con- figuration space {0,1}Λ _{with Λ}

b Z2 into two disjoint sets depending on which circuit around the origin is larger, the 0circuit or the 1∗circuit. To this end, we introduce the following definition.

Definition 4.27 (lasso, ∗lasso) Let ∆be a simply∗connected finite subset of Z2

and fix a node x∈∆. Further, let C be a circuit around x contained in ∆ and let

P be a path starting inC, ending in ∂(∆c_{) and contained in∆. We call the union}

C∪P a lasso around x in ∆.

We extend this definition to0lassos[σ],1lassos[σ],0∗lassos[σ], and so on, where

σstands for the underlying configuration. Most of the times, lassos will be around the origin~0. Therefore, if we omit the phrase “aroundx” we usually mean “around the origin”. Furthermore, we omit “in ∆” if it clear from the context.

Now, we verify a slightly more general relation as the one announced above, namely that a configuration σ ∈ {0,1}Λ _{on a local observation window Λ b} Z2

exhibits either a 1∗lasso as well as a0∗lasso or a 0lasso or a 1lasso.

Lemma 4.28 For all simply ∗connected sets Λ _b Z2 _{with ~0 ∈Λ, the set of con-}

figurations {0,1}Λ _{is a disjoint union of the following three sets}

{σ∈ {0,1}Λ :∃ 1lasso[σ]} {σ∈ {0,1}Λ :∃ 0lasso[σ]}

{σ∈ {0,1}Λ _{:∃ 1∗lasso[σ]∧ ∃ 0∗lasso[σ]}.}

Proof: First, we argue why it is sufficient to show

{0,1}Λ _{={σ∈ {0,1}}Λ _{:∃ 0lasso[σ]} ] {σ ∈ {0,1}}Λ_{:∃ 1∗lasso[σ]}. (4.7)}

Since flipping all spins is bijective, equality (4.7) is equivalent to

{0,1}Λ _{={σ∈ {0,1}}Λ _{:∃ 1lasso[σ]} ] {σ ∈ {0,1}}Λ_{:∃ 0∗lasso[σ]}. (4.8)}

Now Lemma 4.28 follows from the intersection of (4.7) and (4.8), because a lasso is also a∗lasso and it is impossible for a 1lasso and a0lasso to coexist. Therefore, it is sufficient to verify (4.7), which will be done in the sequel.

Note that the origin can be interpreted as a1∗circuit if it takes value 1; other- wise it can be interpreted as a0circuit. Hence, we can always find a 1∗circuit or a

0circuit and compare the size of the maximal1∗circuit to the size of the maximal

0circuit. This is the case because the non-existence of a1∗circuit, i.e.,Cmax 1∗ =∅, implies that even the smallest0circuit – the origin – is larger than every1∗circuit. Further, the absence of a 0circuit implies that the origin has spin value 1 and – interpreted as a∗circuit – is larger than every 0circuit.

There are two types of configurations depending on whether the maximal

1∗circuit is larger than the maximal 0circuit or the other way around. In the first case, by case assumption, the maximal1∗circuit is1∗connected to∂∗(Λc_)and

therefore, is part of a1∗lasso. In the second case, by case assumption, the maximal

0circuit is 0connected to∂(Λc_{) and therefore, is part of a 0lasso. Consequently,} {0,1}Λ _{={σ∈ {0,1}}Λ _{:∃ 0lasso[σ]} ∪ {σ ∈ {0,1}}Λ_{:∃ 1∗lasso[σ]} (4.9)}

holds. Since (Z2,) and (Z2,) are matching pairs, the existence of a 1∗lasso prevents the existence of a 0lasso, i.e.,

{σ∈ {0,1}Λ:∃ 1∗lasso[σ]} ⊂ {σ ∈ {0,1}Λ :_@ 0lasso[σ]}. (4.10) Combining the equations (4.9) and (4.10) yields (4.7). Note that as an immediate consequence a configuration σ ∈ {−1,0,1}Λ _{on a}

local observation window Λ _b Z2 _{exhibits either a 0lasso or a −+∗lasso, i.e., a}

∗lasso in σ−1({−1,1}). This implies that a configuration WR∗(λ)-almost surely exhibits either a 0lasso or a +∗lasso or a −∗lasso in Λ _b Z2_{, i.e., for any µ ∈}

WR∗(λ), the previous property holds µ-almost surely.

Up to now, all statements of this chapter were more or less well-known. This does not apply to the next theorem.

Theorem 4.29 Let λ >0. If

lim sup

Λ%Z2

µ+∗_Λ,λ(∃ 0lasso inΛ)>0,

then phase transition does not occur, i.e.,

|W R∗(λ)|= 1.

Proof: Assume that the condition holds and for contradiction that phase transi- tion occurs.

We note three direct consequences. First, we can fix a sequence of finite subsets

Λn of Z2 containing the origin with Λn%Z2 and an >0 such that

lim sup

n→∞

µ+∗_Λn,λ(∃ 0lasso inΛn)≥ .

Second, due to Lemma 4.21 there exists an infinite +∗cluster µ+∗_λ -almost surely. Third, theµ+∗_λ -almost sure finiteness of all −0clusters follows from Corollary 4.20. Because of the last two statements we can find two integers k, m with k ≤ m

so that withµ+∗_λ -probability at least 1−/2, the infinite+∗cluster hitsΛk and all −0clusters intersectingΛk are contained in Λm. Note that this event is increasing

and implies that for alli≥mwithµ+∗_λ -probability at least1−/2, a+∗lasso inΛi

exists, which is a local increasing event. Thus, for all i≥ m the µ+∗_Λ

i,λ-probability

of the existence of a +∗lasso is at least 1−/2, which leads to

lim sup

n→Z2

µ+∗_Λn,λ(∃ 0lasso inΛn)≤/2,

a contradiction.

Theorem 4.30 Let λ >0. If

lim sup

Λ%Z2

µ+_Λ,λ(∃ 0∗lasso inΛ)>0,

then phase transition does not occur, i.e.,

|W R(λ)|= 1.

The proof of this theorem is almost a copy of the previous one.

4.5

A Condition for the Existence of at Most Two