PERFILES DE COMPETENCIAS INFORMACIÓN GENERAL
REQUISITOS DEL PUESTO DE TRABAJO (1 Base 2 En Desarrollo 3 Avanzado 4 Experto)
We now prove that the notion of P-interface we introduced satisfies the many properties needed in order to carry out the Peierls-type argument sketched at the beginning of this section.
From now on we assume that
Gis an infinite, 1-ended, finitely presented Cayley graph fixed through- out, or more generally, an 1-ended bounded degree graph, admitting a basis P of C(G) whose elements are cycles of bounded lengths (as discussed in Section 10.1).
(29)
We say that a P-interface I = (IV, IO) occurs in a percolation instance ω ∈
{0,1}E(G), if every edge inI
O is occupied and every edge inIV is vacant inω.
We say that I meets a cluster C of ω, if either IO∩E(C) 6= ∅, or IO =
E(C) =∅ andIV =∂C (in which caseC consists ofoonly).
Theorem 10.4. For every finite percolation cluster C of Gsuch that ∂C sep- arates ofrom infinity, there is a unique P-interface(IV, IO)that meets C and
occurs. Moreover, we have IO ⊆E(C)andIV ⊆∂C for thatP-interface.
Conversely, every occurring P-interface meets a unique percolation cluster
C, and∂C separatesofrom infinity (in particular, C is finite).
The proof of this is rather involved, and needs some intermediate steps which we gather now.
The following proposition is based on Timar’s [51] aforementioned proof of the theorem of Babson & Benjamini [10], and contains the quintessence of the notion of aP-interface.
Aminimal cut ofGis a minimal set of edges that disconnectsG. Note that ifB is a minimal cut, thenG−B has exactly two components, and every edge in B has an end-vertex in each of these components.
Proposition 10.5. LetBbe a minimal cut ofGand letL⊂E(G)be a superset of B such that some component D of G−L contains a vertex of each edge in
B. Then B~D is contained in an L-component ofL~D.
Proof. Suppose to the contrary that there are directed edgese, f ∈B~D that lie
in distinctL-components ofL~D. Note thate, f cannot be the two directions of
the same undirected edge because no edge ofB has both end-vertices inD by the above remark about minimal cuts. Let (L1, L2) be a proper bipartition of
~
LD such thate∈ L
1, f ∈L2, and there is no P-path in G−L connectingL1 toL2, which exists by the definitions and the fact thatL~Dis partitioned by its
L-components by (27).
LetR be an e-f path in D, which exists becauseD is assumed to contain a vertex of each edge in B. LetQbe ane-f path in the component of G−B
avoidingD; this component exists becauseG−B has exactly two components, one of which containsD sinceL⊇B (Figure 5).
LetKbe the cycle obtained by joining these pathsR, Qusingeandf. Since
P is a basis for the cycle spaceC(G), we can expressKas a sumPC
iof cycles
Ci∈ P, where this sum is understood as taking place inC(G).
Note that no cycleCi contains a path inG−LconnectingL1toL2, because no such path exists by the choice of (L1, L2). Let LCi :=
←−−−−−→
L∩E(C) be the directions of edges ofLappearing inCi. The previous remark implies thatLCi
has an even number of its elements in each ofL1, L2, because each component ofCi−L(which is a subpath ofCi) is incident with either 0 or 2 such elements
pointing towards the component, and they lie both inL1 or both inL2or both in none of the two.
R Q e B f L Ci D
Figure 5: The situation in the proof of Proposition 10.5.
This leads into a contradiction by a parity argument: notice that our cycle
K contains an odd number of directions of edges in each of L1, L2, namely exactly one in each —eandf respectively— becauseP avoidsLand Qavoids
D, hence L~D, by definition. But then our equality K = PC
i is impossible
by the above claim because sums inC(G) preserve the parity of the number of (directed) edges in any set. This contradiction proves our statement.
We can use the same ideas to prove the following proposition.
Proposition 10.6. Let L ⊆ E(G), let D be a component of G−L, and let
e = vz be an edge of L such that v, z ∈ V(D). Then vz, ~~ zv lie in the same
L-component of L~D.
Proof. It is not hard to adapt the proof of Proposition 10.5 to our setup to prove our statement; the only difference is that instead of the cycleK we now consider a cycle consisting of the edge vzand av–z path inD. But we can in fact just apply Proposition 10.5 to an auxiliary graph to deduce Proposition 10.6 as follows. Subdivide the edgevzinto two edgesvw, wzby adding a new vertex
w. Consider the minimal cutB of the resulting graph that consists of these two edgesvw, wz (and separateswfrom the rest of G). Applying Proposition 10.5 to this graph after replacing L with L0 := L−vz∪ {vw, wz} we deduce that
~
wz, ~wvlie in the sameL0-component ofL~0D, and it is straightforward to deduce
that vz, ~~ zvlie in the sameL-component ofL~D from this.
Next, we prove one of the desired properties of P-interfaces, namely that
IV ∪IO spans a connected subgraph ofG.
Proposition 10.7. For every P-interface I = (IV, IO) of G, the edge-set IO
spans a connected subgraph of G incident with all edges in IV, unless IO =∅
(in which case IV is the set of edges incident with o).
Proof. LetDbe defined as in (ii) of Definition 10.3. By (iv) of Definition 10.3, for everye∈IO there is aP-pathP in G−IV connectinge to the head of an
element ofI~D
V. Note that all edges ofP belong toIO as we can apply item (iv)
to any of them, where we use the fact that sinceP meets D, it is contained in
D because D is a component ofG−IV. This means that every component of
the graphGO⊆Gspanned by the edges inIO contains the head of an element
ofI~D V.
Therefore, if GO has more than one components, then these components
define a proper bipartition (J1, J2) ofI~VD, by letting J1 be the set of allj∈I~VD
such thathead(j) lies in one of these components. Applying Definition 10.1 to this bipartition we obtain a contradiction, since for any P-path P in G−IV
connecting j1 ∈ J1 to j2 ∈ J2, all edges of P lie inIO by the above remark,
which implies that the heads of j1 and j2 lie in the same component of GO.
This proves thatGO is connected as claimed.
Finally, if somee∈IV is not incident withGO, then we can apply the same
argument to the bipartition of I~D
V one partition class of which consists of the
one or two directions of e that lie in I~D
V (recall the remark after (iii) of Defi-
nition 10.3). IfIO6=∅, then this bipartition is proper because each component
of GO is incident with an element of I~VD as we have proved, and we obtain a
contradiction as above.
IfIO =∅, and there are at least two verticesx, yofDincident withIV, then
we obtain a proper bipartition of IV by letting one of the classes be the set of
edges incident withx, say, and reach a contradiction with the same arguments. Thus all edges ofIV are incident with a vertexxofD in this case, and in order
to satisfy (i)IV must be the set of edges incident withx=o.
We have now gathered enough tools to prove our main result about P- interfaces.
Proof of Theorem 10.4. Existence: To begin with, given such a clusterC we will find an occurring P-interface (IV, IO) such that IO ⊆E(C) and IV ⊆∂C.
For this, let
B:={e∈∂C| there is a path frometo ∞inG−∂C}.
(This is the minimal subset of∂C separating Cfrom infinity.)
Fix an enumeration of the elements of B~C (this notation was introduced
before Definition 10.3), and let Xi,1 ≤ i ≤ |B~C| be the ∂C-component of ↔
∂C containing the ith element of B~C in that enumeration (the definition of
F-components is given after Definition 10.1). It will turn out that these com- ponents Xi coincide with each other, but we cannot use this fact yet. Let
J := S
iXi, and let IV be the corresponding undirected edges, that is, IV :=
{vw∈∂C|vw~ ∈J}.
We will start by proving thatIV satisfies properties (i), (ii) and (iii), after
which we can define IO via (iv) to ensure that (IV, IO) is indeed aP-interface.
To see that (i) is satisfied, we recall thatB⊆IV by the definitions, and we
claim that B separates o from infinity. This is true because ifQis an infinite path starting at o, then it has to contain an edge in ∂C by our assumption that ∂C separates o from infinity. The last such edge of Q then lies in B by the definitions. Thus all paths from o to infinity meet B, proving that (i) is satisfied.
It is easy to see that (ii) is satisfied by lettingDbe the component ofG−IV
containing C, which exists since IV ⊆∂C. Indeed, C ⊆D meets all edges in
We will now check that I~D
V is IV-connected, that is, (iii) is satisfied. Pro-
position 10.5 —applied with L=IV, so that D meets all edges in B ⊆IV as
remarked above— yields thatB~D is contained in someI
V-componentX ofI~VD.
We will prove thatX contains the other edges of I~D
V too. For this, recall that
Xi is a ∂C-component of ↔
∂C, and so Xi is ∂C-connected by the definition of
∂C-components. We can reformulate this by saying that Xi is (contained in)
a ∂C-component of Xi. Recall thatJ = SiXi. Using (28) with F =∂C we
will show thatJ ⊆I~D
V. Indeed, the componentCofG−∂C contains the head
of an element of Xi in B~C by the definition of Xi, and so the head of every
element ofJ lies inCby (28). SinceC⊆D, we deduceJ ⊆I~D
V. Plugging these
facts into Proposition 10.2 —withY =Xi— we obtain thatXi is contained in
anIV-component ofI~VD, becauseXi ⊆I~VD andIV ⊆∂C. Since eachXi meets
~
BD, which is contained in the I
V-component X, (27) yields that X contains
J =S iXi.
To conclude that I~D
V is IV-connected, or in other words, that X = I~VD, it
remains to show that ife∈I~D
V −J thenelies inX as well. To see this, note that
for any such e=vz~ the reverse direction e0 :=zv~ lies in J, because all edges of IV have at least one of their directions inJ by the definitions. Moreover,
we have z, v ∈ V(D) since e, e0 ∈ I~D
V, where we used the fact that J ⊆ I~VD.
Thus Proposition 10.6 —with L =IV— yields that e, e0 lie in a common IV-
component ofI~D
V. Using (27) again, combined with the fact that (e0 ∈)J ⊆X
proved above, we deduce thate∈X as desired. To summarize, we have proved that all elements of I~D
V lie in a common IV-componentX, in other words, I~VD
isIV-connected, establishing (iii).
We proved above that J ⊆ I~D
V. Next, we claim that actually
~ ID
V = J,
which will be used below. Suppose this is not the case, and consider the proper bipartition (J, ~ID V −J) of ~ ID V. Since ~ ID
V isIV-connected, there is aP-pathP in
G−IV connecting directed edges e∈J tof ∈I~VD−J. Let gbe the first edge
of P that lies in ∂C, directed towardse, if such an edge exists, and let g =f
otherwise. In both cases, the subpathP0 ofP frometogavoids∂C, and hence proves thateandglie in a common∂C-component of
↔
∂C. But thengmust lie in J since J is a union of∂C-components of
↔
∂C. This contradicts thatg 6∈J
wheng=f andg6∈IV otherwise. This contradiction proves thatI~VD=J.
Thus using (iv) of Definition 10.3 to define IO, we obtain a P-interface
I:= (IV, IO). SinceIV ⊆∂C which is vacant, to show thatIoccurs it remains
to show that IO is occupied in ω. This is true because if P is a P-path in
G−IV connecting some edge e of IO to I~VD = J, then the last vacant edge
f of the extended path {e} ∪P, if such an edge f exists, would have to lie in IV by the definitions and the fact thatI~VD =J, contradicting that {e} ∪P
avoids IV. Hence no suchf exists, and in particular anye∈IO is occupied as
desired. Moreover, I meets C because IO∪IV spans a connected subgraph of
Gby Proposition 10.7, and that subgraph containsB, hence meetsC.
To prove the claim thatIO ⊆E(C), recall that IO spans a connected sub-
becauseGo is incident with all ofIV ⊇B, and it cannot meet the infinite com-
ponent ofG−B as it is contained inD. Since IO, being occupied, avoids ∂C,
we deduce thatIO⊆E(C) indeed.
Uniqueness: Suppose that our clusterC is met by a further occurringP- interface I0 = (IV0 , IO0 ) 6= I. By Lemma 10.7 the subgraph of G spanned by
IO0 ∪IV0 is connected, and therefore contained inC∪∂C sinceIO0 meetsE(C). It follows thatIV0 ⊆∂C sinceI0 occurs.
LetD0 be the component ofG−IV0 defined in (ii). We claim thatB ⊂IV0 . Indeed, if IV0 misses some edge of B, thenIV0 ⊆∂C does not separate C from infinity, hence C∩D0=∅, contradicting that IO0 ⊆E(D0) andIO0 ∩E(C)6=∅
unlessE(C) =∅, in which caseIV0 cannot separateofrom infinity violating (i). Moreover, we haveD0⊇CsinceI0
V ⊆∂C (becauseI0 occurs) andIO0 meets
E(C).
We will first prove thatIV0 ⊆IV. So letf ∈IV0 , and suppose for a contra-
diction thatf 6∈IV. In this case, the bipartition (J, J0 := ↔
∂C−J) of
↔
∂C, where
J is as in the definition of IV in the existence part, is such that B~C ⊆J and
both directions f ,~ ~f off lie inJ0 and there is noP-path inG−∂C connecting
J toJ0.
Consider now the bipartition (J ∩I~0D0
V , J0∩I~0D 0 V ) of I~0D 0 V , which is proper because B~C ⊆I~0D0
V (because D0 ⊇ C and B ⊂IV0 ) and{f ,~ } ∩~f I~0D
0
V 6=∅ (by
the definition of D0). Therefore, since I~0D0
V isIV0 -connected by (iii), there is a
P-pathP inG−IV0 connectingJ∩I~0D0
V toJ0∩I0~D
0
V . Letebe the last edge ofP
in∂C, which exists becauseP cannot avoid∂C by the aforementioned property of the bipartition (J, J0), and letP0be the final subpath ofP starting ate. But then applying (iv) toI0 using the pathP0 we deduce thate∈IO0 , contradicting that I0 occurs and e∈∂C is vacant. This contradiction proves thatIV0 ⊆IV.
Next, we prove thatIV ⊆IV0 as well. Indeed, ifIV 6⊆IV0 , then the bipartition
(I~D V ∩ ↔ IV0 , ~ID V − ↔ IV0 ) of I~D V is proper because B~C ⊆ I~VD, ↔ IV0 . Since I~D V is IV-
connected, there is aP-pathP inG−IV connecting some edgef ∈IV0 to some
edgee∈IV −IV0 . Since we have proved thatIV0 ⊆IV, we deduce thatP lies in
G−IV0 . But then applying (iv) to I0 using the pathP we deduce thate∈IO0 , contradicting that I0 occurs and e ∈ IV ⊆ ∂C is vacant. This contradiction
proves thatIV ⊆IV0 , and henceIV0 =IV.
To conclude that I is the unique occuring P-interface that meets C, it re- mains to prove thatIO0 =IO. But this is now obvious from (iv), sinceIV0 =IV
and henceD0=Dby (ii).
Converse: Suppose now that (IV, IO) is aP-interface occurring in a perco-
lation instanceω. Then by Lemma 10.7 it meets a unique cluster C ofω, and we haveIV ⊆∂C by what we proved above. By (i)IV, and hence∂C, separates
ofrom infinity.