Proof of Proposition 5.8

It may help the reader to refresh the core idea of this subsection, see page 77. Let Λ _b Z2 be a simply ∗connected set with ~0 ∈ Λ. We only prove the statement of Proposition 5.8 for this fixed set Λ. Sometimes, we even omit the index Λ, e.g.,m denotes the mapmΛ in the sequel.

Recall that the domain A1 _{of the map m is defined as}

{∃ 0lasso, ~0∈/ C_Λmax 0, ∂∗C_Λmax 0

1∗

6

←→∂∗C_Λmax 1}.

The existence of a 0lasso implies that the maximal 0circuit, which is weakly

0connected to Λc, is larger than every 1circuit (if a 1circuit exists at all), where by saying is weakly 0connected to Λc _{we mean that its boundary is0connected to}

∂(Λc_{). Moreover, one can find a second 0circuit in intC}max 0

Λ again larger than any

1circuit, since the origin is not contained in the maximal0circuit and the maximal

0circuit is not ∗weakly1∗connected to the maximal1circuit. This implies (in fact it is equivalent) that the maximal 0circuit in intCmax 0

Λ exists and is 0∗connected

to C_Λmax 0. Summing up, we can state

A1 ={∃0lasso, ~0∈/ C_Λmax 0, ∂∗C_Λmax 0

1∗

6

←→∂∗C_Λmax 1, C_Λmax 0 > C_Λmax 1,∅ 6=C_intmax 0_Cmax 0

Λ

0∗

←→C_Λmax 0, C_intmax 0_Cmax 0 Λ > C

max 1

Λ } (5.5)

Figure 5.1 illustrates the properties of a configuration of A1_.

After analysing the domain, we state some useful fundamental relations be- tween σ∈A1 _{and m(σ).}

For this task recall the definition of the circuit

i_Cfill_{(σ) = min}i_CCmax 0

Λ (σ)∪σ

−1_(1)∩

extC_intmax 0_Cmax 0 Λ (σ)(σ)

(5.6) for σ ∈ A1_{, see page 75. In other words, this circuit,} i_Cfill_{(σ), is the minimal}

induced circuit that satisfies the following two conditions:

• It is contained in the union of the maximal 0circuit[σ], Cmax 0

Λ (σ), and the

set of nodes equipped with 1spins[σ], σ−1(1);

• It is strictly larger than the “second largest“ 0circuit[σ], Cmax 0 intCmax 0

Λ (σ)

(σ). Note that this description equates our definition.



 0

Figure 5.1: In this figure the white squares represent nodes with 0spins[σ, m(σ)]

and the black squares are nodes with1spins[σ, m(σ)]. The gray squares are nodes equipped with 0spins[σ] and 1spins[m(σ)]. The maximal 0circuit[σ] is indicated by a blue curve and consists of white and gray squares. The circuit iCfill(σ) is indicated by a red curve and consists of black and gray squares. The “second largest” 0circuit[σ],Cmax 0

intCmax 0 Λ (σ)

(σ), is indicated by a green curve.

Remark 5.15 Let σ ∈A1. Then the following properties hold

σ−1(1)∪C_Λmax 0(σ)⊃m(σ)−1(1) (5.7) σ−1(0) =m(σ)−1(0)∪C_Λmax 0(σ) (5.8) i_Cfill_(σ)∩Cmax 0 Λ (σ)6=∅ (5.9) ∂∗iCfill(σ)σ −1_(1)∗ ←→ ∂∗(Λc) (5.10)

C_intmax 0_Cmax 0

Λ (σ)(σ)⊂σ −1

(0)∩m(σ)−1(0) (5.11)

C_Λmax 0(σ)≥C_extmin 1_Cmax 0

Proof: Fix a configurationσ ∈A1. By the definitions of both the configuration

m(σ) = ₁σ−1_(1)∪i_Cfill_(σ) (5.13)

and the circuit

i_Cfill_{(σ) = min}i_CCmax 0

Λ (σ)∪σ −1

(1)∩extC_intmax 0_Cmax 0 Λ (σ)(σ)

,

the statements (5.7) and (5.8) are evident.

For the next properties let us first define the ”half-open” respectively ”open” annulus specified by the maximal 0circuit[σ] and the ”second largest“ 0circuit[σ]

by

i

C_intmax 0_Cmax 0

Λ (σ)(σ), C max 0 Λ (σ)

i

:=C_Λmax 0(σ)∪intC_Λmax 0(σ)∩extC_intmax 0_Cmax 0 Λ (σ)(σ)

respectively

i

C_intmax 0_Cmax 0

Λ (σ)(σ), C max 0 Λ (σ)

h

:=intC_Λmax 0(σ)∩extC_intmax 0_Cmax 0

Λ (σ)(σ).

Next, recall the description of i_Cfill_{(σ) after (5.6). In particular, the second con-}

dition of this description said that the circuit iCfill(σ) is strictly larger than the "second largest"0circuit[σ],Cmax 0

intCmax 0 Λ (σ)

(σ). Moreover, the first condition, together with minimality of i_Cfill_{(σ), immediately implies that} i_Cfill_{(σ) is smaller than the}

maximal 0circuit[σ], C_Λmax 0(σ). Summing up, we know that iCfill(σ) lies in the ”half-open” annulus, i.e.,

i_Cfill_(σ)⊂i_Cmax 0 intCmax 0 Λ (σ)(σ), C max 0 Λ (σ) i . (5.14)

Further, by definition, we know that all nodes of iCfill_{(σ) not contained in the}

maximal 0circuit[σ] are equipped with 1spins[σ], i.e.,

i_Cfill_(σ)∩(Cmax 0 Λ (σ))

c_⊂σ−1

(1). (5.15)

In other words, a node ofi_Cfill_{(σ)has1spin[σ]if and only if it belongs to the ”open”}

annulus, in short for all x∈i_Cfill_(σ)

σ(x) = 1 ⇐⇒ x∈iC_intmax 0_Cmax 0

Λ (σ)(σ), C max 0 Λ (σ)

h

.

Property (5.9) is a consequence of these two Observations (5.14) and (5.15), together with the choice of

σ ∈A1 (5.5)⊂ {π∈ {0,1}Λ_:Cmax 0 intCmax 0

Λ (π)(π) 0∗

i.e., the maximal 0circuit[σ] and the "second largest" 0circuit[σ] are 0∗connected. Since every node of Cmax 0

Λ (σ) is ∗weakly 1∗connected[σ] to Λc, Property (5.10)

follows from Property (5.9).

As before iCfill(σ) is strictly larger than the "second largest" 0circuit[σ], see (5.14). Hence, since we only change spin values in iCfill_{(σ), see (5.3), inclusion}

(5.11) follows.

The last Statement (5.12) is a direct consequence of the facts that iCfill(σ) is smaller than the maximal0circuit[σ], see (5.14), and thatiCfill_{(σ)is a1circuit[m(σ)]}

∗weakly 1∗connected[m(σ)] toΛc_{, see (5.13) and (5.10).}

These remarks enable us to describe the image of m|_A1.

Lemma 5.16 Let σ ∈A1. Then mΛ(σ) is an element of

{∃ 1∗lasso, ~0∈/C_Λmax 1 6=∅, C_Λmax 1

1∗

6←→C_intmax 1_Cmax 1

Λ }.

Proof: Let σ∈A1 and recall that

i_Cfill_{(σ) = min}i_CCmax 0

Λ (σ)∪σ −1

(1)∩extC_intmax 0_Cmax 0 Λ (σ)(σ)

.

A direct consequence of the definition of i_Cfill_{(σ) is that the ∗boundary of every}

node ofiCfill(σ)is0∗connected[σ, m(σ)]in(iCfill(σ))c toC_intmax 0_Cmax 0

Λ (σ)(σ), i.e., for all

nodes z ∈i_Cfill_(σ)

∂∗z ←→0∗ C_intmax 0_Cmax 0

Λ (σ)(σ) in(

i_Cfill_(σ))c

(5.16) holds.

Since the circuitiCfill(σ)is a 1circuit[m(σ)], see (5.4), which is ∗weakly 1∗con- nected[m(σ)] to Λc_{, see (5.9) and (5.4), the existence of a 1∗lasso[m(σ)] follows,}

i.e.,

m(σ)∈ {∃ 1∗lasso}.

But we have already verified the existence of a 0circuit[m(σ)], e.g.Cmax 0 intCmax 0

Λ (σ)

(σ), see (5.10), which, therefore, has to be smaller than the1circuit[m(σ)]i_Cfill_(σ)that

is∗weakly 1∗connected[m(σ)]toΛc. Consequently, the origin cannot be contained in the maximal1circuit[m(σ)], i.e.,

m(σ)∈ {~0∈/ C_Λmax 16=∅}.

Recall that the circuit iCfill_{(σ) intersects the maximal 0circuit[σ], see (5.9),}

which is0connected[σ]to the boundary ofΛc_{, see (5.5), and that the configuration}

σ equals m(σ) outside of iCfill(σ). Hence, there exists a node y∈i_Cfill_{(σ) with}

i.e., we can find a nodey∈i_Cfill_{(σ)that is weakly0connected[σ, m(σ)]toΛ}c_{. This,}

together with (5.16), implies that y∈i_Cfill_{(σ) satisfies}

∂(Λc)←→0 ∂∗y←→0∗ C_intmax 0_Cmax 0

Λ (σ)(σ)in ( i

Cfill(σ))c. (5.17) Thus, one cannot find two disjoint1circuits[m(σ)]in extCmax 0

intCmax 0 Λ (σ)

(σ),since both would have to intersect the node y. This, together with the fact that i_Cfill_{(σ) is a}

1circuit[m(σ)], see (5.4), strictly larger than the 0circuit[σ, m(σ)] C_intmax 0_Cmax 0 Λ (σ)(σ),

see (5.11), implies

m(σ)∈ {C_Λmax 1

1∗

6←→C_intmax 1_Cmax 1

Λ },

which concludes the proof.

It may help the reader to – once again – refresh the core idea of this subsection described on page 77. Our next step is to “connect” a configuration m(σ) to the original configuration σ∈A1. More precisely, we want to determine both i_Cfill_(σ)

from m(σ)∈m(A1)and

i

Cempty(m(σ)) := maxiC

C_extmin 1_Cmax 0

Λ (m(σ))(m(σ))∪m(σ) −1

(0)

= maxiCiC_extmin 1_Cmax 0

Λ (m(σ))(m(σ))∪m(σ) −1

(0) (5.18) from σ∈A1, where the identity follows from

i_Cmin 1 extCmax 0 Λ (m(σ))(m(σ))≥C min 1 extCmax 0 Λ (m(σ))(m(σ)) and i_Cmin 1 extCmax 0 Λ (m(σ))(m(σ))⊂C min 1 extCmax 0 Λ (m(σ))(m(σ)).

Little is known, but i_Cempty_{(m(σ)) is the circuit that changes m(σ) into σ if it is}

emptied.

Lemma 5.17 Let σ ∈ A1. Then the minimal induced 1circuit[m(σ)] in the exte- rior of the maximal 0circuit[m(σ)], iCmin 1

extCmax 0 Λ (m(σ))

(m(σ)), equalsiCfill_{(σ) and the}

maximal induced 0circuit[σ] iC_Λmax 0(σ) equals i_Cempty_{(m(σ)), i.e.,} i_Cmin 1

extC_Λmax 0(m(σ))(m(σ)) =

i_Cfill_{(σ) (5.19)} i_Cempty_{(m(σ)) =}i_Cmax 0

Λ (σ). (5.20)

Proof: Since iCfill(σ) is a 1circuit[m(σ)], see (5.4), that is ∗weakly 1∗connect- ed[σ, m(σ)]toΛc_{, see additionally (5.10), we know thatC}max 0_{(m(σ))⊂int}i_Cfill_(σ),

which, together with (5.4), leads to

i_Cmin 1 extCmax 0

Λ (m(σ))(m(σ))≤

On the other hand

i

Cfill(σ) = miniC

C_Λmax 0(σ)∪σ−1(1)∩extC_intmax 0_Cmax 0 Λ (σ)(σ)

= miniC(C_Λmax 0(σ)∪σ−1(1))∩extC_intmax 0_Cmax 0 Λ (σ)(σ)

(5.11)

≤miniC (C_Λmax 0(σ)∪σ−1(1))∩extC_Λmax 0(m(σ))

(5.7)

≤miniC m(σ)−1(1)∩extC_Λmax 0(m(σ)) =iC_extmin 1_Cmax 0

Λ (m(σ))(m(σ)).

Both inequalities together give (5.19). This in turn implies the following:

m(σ)−1(1) (5.4)= σ−1(1)∪i_Cfill_(m(σ)) (5.19) ⇐⇒ m(σ)−1(0) =σ−1(0)\i_Cmin 1 extCmax 0 Λ (m(σ))(m(σ)), which verifies

σ−1(0)⊂m(σ)−1(0)∪C_extmin 1_Cmax 0

Λ (m(σ))(m(σ)). (5.21)

On the one hand

i_Cempty_{(m(σ)) = max}i_CCmin 1 extCmax 0 Λ (m(σ))(m(σ))∪m(σ) −1 (0) (5.21) ≥maxiC σ−1(0) =iC_Λmax 0(σ)

and on the other hand

i_Cempty_{(m(σ)) = max}i_CCmin 1 extCmax 0 Λ (m(σ))(m(σ))∪m(σ) −1 (0) (5.12) ≤maxiC C_Λmax 0(σ)∪m(σ)−1(0)) (5.8) =iC_Λmax 0(σ).

Considering both inequalities yields (5.20). Having established the above “connection” between σ and m(σ), we are ready to see that

m−1 :m(A1)→! A1; m(σ)7→1−_1m(σ)−1_(0)∪i_Cempty_(m(σ))

!

=σ

is a well-defined promising candidate for the inverse map, which is illustrated in Figure 5.1.

Lemma 5.18 Let σ ∈ A1. Then 1− ₁m(σ)−1_(0)∪i_Cempty_(m(σ)) = σ, i.e., the map

mΛ|A1 is injective.

Proof: It is sufficient to show

i_Cfill_(σ)∩σ−1

(0)=! iCempty(m(σ))∩m(σ)−1(1)

if we want to prove 1−₁m(σ)−1_(0)∪i_Cempty_(m(σ))=σ.

"⊂" This inclusion is a consequence of

i_Cfill_(σ)(5.19)₌ i_Cmin 1 extCmax 0 Λ (m(σ))(m(σ))⊂m(σ) −1₍₁₎ and i

Cfill(σ)∩σ−1(0)(5.1)⊂ iC_Λmax 0(σ)(5.20)= iCempty(m(σ)).

"⊃" This inclusion is a consequence of

i Cempty(m(σ))(5.20)= iC_Λmax 0(σ)⊂σ−1(0) and i_Cempty_{(m(σ))∩m(σ)}−1 (1)(5.18)⊂ i_Cmin 1 extCmax 0 Λ (m(σ))(m(σ)) (5.19) = iCfill(σ),

In document Carstens, Sebastian Maurice (2012): Percolation analysis of the two-dimensional Widom-Rowlinson lattice model. Dissertation, LMU München: Fakultät für Mathematik, Informatik und Statistik (Page 93-100)