Zigzag diagrams and Martin boundary

0.37 MB·English

by  Pierre Tarrago

ZIGZAG DIAGRAMS AND MARTIN BOUNDARY
PIERRE TARRAGO
5
1
0
2

n
Abstract. In this paper, we investigate the Martinboundary of a gradedgraph
a
J whose paths represent coherent sequences of permutations. This graph was
8 Zintroduced by Viennot and its boundary studied by Gnedin and Olshanski. We
2 provethattheMartinboundaryofthisgraphcoincideswithitsminimalboundary.
Then we relate paths on this graph with paths on the Young graph, and finally

]
give a central limit theorem for the Plancherel measure on the paths of .
R
Z
P
.
h
1. Introduction
t
a
m The lattice of zigzag diagrams is a graded graph whose vertices of degree n
Z
[ are labelled by compositions of n (which can be seen as ribbon Young diagrams).

1 The study of this kind of lattices drew increasing interests these last decades, due
v to their interactions with representations of semi-simple algebras and with discrete
7
random walks. In particular an other example of graded graph, the Young lattice ,
8
Y
0 has been deeply studied by Vershik, Kerov and other authors (see [11] for a review
7
on the subject), yielding major breakthroughs on the representation theory of S
0
∞
. and on the asymptotic study of certain particle systems. As explained in [8], the
1
0 lattices and are somehow related, since the latter can be seen as a projection
Z Y
5 of the former.
1
The connection between the lattice structure and its probabilistic applications is
:
v
made through the study of harmonic functions on the associated graph. One of the
i
X first tasks is therefore the characterization of harmonic functions on the lattice ; it
r is then possible to single out particular harmonic functions and study the random
a
variables they generate. A general framework for the representation of harmonic
functions on a graph E was initiated by Martin in [15] with the concept of Martin
boundary ∂ E and minimal boundary ∂E : the Martin boundary is a topological
M min
space associated to the graph that allows to establish a bijection between positive
harmonic functions and measures whose support is included in a particular subset
of ∂ E. The latter subset is precisely the minimal boundary ∂E . It is there-
M min
fore important to know both ∂ E and ∂E to provide a topological and measure
M min
theoretic approach to the kernel representation of harmonic functions (see [4] for an
exhaustive review on the subject).
In general ∂E is strictly included in ∂ E. However in many cases the two coin-
min M
cide, as it happens for example for the lattice . In this paper we prove that the
Y
two boundaries also coincide for the lattice . The minimal boundary of has
Z Z
1
2 PIERRETARRAGO
already been described by Gnedin and Olshanski in[8], through the so-called ori-
ented paintbox construction, and thus it remains to prove that any element of the
Martin boundary fits in this construction. As an application we provide a precise
link between harmonic measures on and harmonic measures on : this link was
Y Z
already exposed in [8], and in the present article we explain this relation by mapping
directly paths on to paths on . Finally we study the behavior of a random path
Z Y
with respect to the Plancherel measure by providing a Central Limit Theorem.
Section 2 and 3 are devoted to preliminaries : the first gives necessary backgounds
on Martin boundary, and the second describes the graph together with its link
Z
with compositions. The results of Gnedin and Olshanski on this graph are given in
Section 4. In this section we provide also the pattern of the proof for the identifica-
tion of the Martin boundary.
The proof heavily relies on combinatorics of compositions. In particular the Martin
kernel of , a two parameters function that characterizes the Martin boundary, is
Z
related to standard fillings of compositions. Two constructions are needed in order
to identify the Martin boundary: Section 5 deals with the first one, which is the
construction of a sequence of random variables that relates the Martin kernel to
the oriented Paintbox construction of Gnedin and Olshanski. The second one has
been done in [19] and is a general framework that gives combinatoric estimates on
compositions. Some results of [19] are recalled in Section 6. Section 7 and 8 show
the identification of the Martin boundary. Finally Section 9 gives the map between
paths on and paths on and exposes probabilistic results associated to a particu-
Z Y
lar point of the Martin boundary, called the Plancherel measure (due to its relations
with the Plancherel measure on the graph ).
Y
We should stress that the map between the paths on the two graphs appears clearly
by using the algebra FQSym of Free Quasi-Symmetric functions; although this al-
gebra won’t be explained in this paper, the interested reader should refer to the
Chapter 3 of [5] for an introduction to FQSym and an explanation of the construc-
tion we are doing in Section 9.2 of the present paper.
2. Graded graphs and Martin boundary
This section is a discussion that introduces the concept of Martin boundary and
motivates its role in the framework of graded graphs. All these results and proofs
can be found in [4].
2.1. Graded graphs and random walk. The notations used here are from [18].
A rooted graded graph is the data of a triple (V,ρ,E) where :
G
V is a denumerable set of vertices with a distinguished element µ .
0
•
ρ : V N is a rank function with ρ 1( 0 ) = µ .
− 0
• → { } { }
The adjacency matrix E is a V V-matrix with entries in 0,1 , such that
• × { }
E(µ,ν) is zero if ρ(ν) = ρ(µ)+1.
6
ZIGZAG DIAGRAMS AND MARTIN BOUNDARY 3
We write µ ν if E(µ,ν) = 1. A path on is sequence of vertices (µ ,...,µ ,...)
1 n
ր G
of increasing degree such that for all i 1, µ µ . For a given graded graph
i i+1
≥ ր
the paths counting function d : N is the function that gives for each vertex
∗
V →
µ the number of paths between µ and µ. When the endpoints of a path are
0
∈ G
not specified, the path is considered as an infinite path starting at the root.
There is a natural way of constructing random walks that respect the structure of
the graph : such a random walk starts at µ , and at each step the successor is
0
G
chosen according to a transition matrix P, with the condition that P(µ,ν) = 0 if
E(µ,ν) = 0 and P(µ,ν) = 1.
ν
Thus each transition matrix P associates to any path λ = (µ µ µ ) a
0 1 n
P ր ··· ր
weight p which is the probability of the realization of λ, namely
λ
1
p = P(X = µ ,X = µ ,...,X = µ ) = P(µ ,µ )...P(µ ,µ ).
λ 0 0 1 1 n n 0 1 n 1 n
Z −
ForsometransitionmatricesP on , theweightp(λ)onlydependsonthefinalvertex
G
of the path (in this case we write p(λ) = p(µ) for any path λ between µ and µ);
0
such a transition matrix is called an harmonic matrix. In this case, a staightforward
computation shows that p, the associated weight function, must verify
(1) p(µ) = p(ν),
µ ν
Xր
and conversely, any positive solution p of (1) such that p(µ ) = 1 yields an harmonic
0
matrix. This can be interpreted in terms of potential theory.
Let X be a denumerable states space with transition matrix P. Let H(X,P)+ (resp.
M(X,P)) denote the set of positive harmonic functions (resp. positive harmonic
measures), which is the set of functions f : X R+ satisfying P(x,y)f(y) =
→ y
f(x) (resp. f(x)P(x,y) = f(y)). For each α M(X,P) let the dual transition
x ∈ P
matrix Pt be defined by the expression
α
P
α(y)
Pt(x,y) = 1 P(y,x),
α α(x)6=0α(x)
if x = y, and Pt(x,x) = 1 Pt(x,y). Then Pt is indeed a transition matrix
6 − x=y α
on X and the following maps are6 well-defined:
P
H(X,P)+ M(X,Pt)
H : → α ,
α h (x 1 1 h(x))
(cid:26) 7→ 7→ α(x)>0α(x)
and
M(X,P) H(X,Pt)+
M : → α .
α m (x α(x)m(x))
(cid:26) 7→ 7→
The two maps are bijective if α > 0 on X.
Let P be a transition matrix on a graded graph ; by a recursive computation there
G
exists aunique invariantmeasureα withrespect toP suchthatα (µ ) = 1. IfP is
P P 0 p
4 PIERRETARRAGO
an harmonic matrix, with p the associated weight function, then P(µ,ν) = 1 p(ν)
µ νp(µ)
ր
and α = d(µ)p(µ). Thus the dual transition matrix is
p
d(µ)p(µ)p(ν) d(µ)
Pt (ν,µ) = 1 = 1 .
αp µրνd(ν)p(ν)p(µ) µրνd(ν)
In particular Pt is independent of p and, by H , any harmonic function of P comes
αp αp
from an invariant measure of Pt. Conversely let α be an invariant measure of Pt.
Then the dual matrix (Pt)t is exactly P , the harmonic matrix associated to the
α α/d
weight function p = α/d. We can check that the duality yields indeed a bijection
between harmonic matrices of and elements of M( ,Pt) taking the value 1 on µ .
0
G G
Thus the problem of finding the harmonic matrices on is equivalent to the dual
G
problem of finding harmonic measures with respect to Pt. Moreover an answer to
the latter problem gives also by duality all the harmonic functions with respect to
an harmonic matrix.
Fortunately a general framework, the Martin entrance boundary, describes exactly
the harmonic measures associated to a transition matrix.
2.2. Martin entrance boundary. Let us take a closer look at the Markov chain
( ,Pt). Let n 1 and ν a vertex of degree n . The random walk X = (X )
0 0 n n 0
wGith transition m≥atrix Pt and initial distribution δ goes backward from ν to µ an≥d
ν 0
stops at µ at the times n . Let λ be a path between µ and ν; from the definition
0 0 0
of the kernel Pt, the probability that X follows the path λ is independent of λ and
is therefore 1 .
d(ν)
For µ of degree m n , denote by d(µ,ν) the number of paths between µ and ν
0
≤
(and by extension d(µ,ν) = 0 if the degree of µ is larger than the one of ν). By
counting the paths going from µ to ν and passing through µ, the probability that
0
X = µ is thus
n0 m
− d(µ)d(µ,ν)
P(X = µ) = .
n0−m d(ν)
In particular setting α (µ) = d(µ)d(µ,ν) yields a measure α that is harmonic with
ν d(ν) ν
respect to Pt, except on the vertex ν. To construct actual harmonic measures, it
seems thus natural to look at the behavior of α when ν . Making the latter
ν
→ ∞
rigorousrequirestospecifyaconvergencemodeforsequences ofvertices ofincreasing
degree. Let K (ν) = d(µ,ν) be the Martin kernel of , and define on the metric :
µ d(ν) G G
1
D(ν ,ν ) = K (ν ) K (ν ) ,
1 2 2Γ(µ)| µ 1 − µ 2 |
µ
X
Γ being any injective function V N. Through this metric, V is seen as a subset
→
of the space of functions from V to [0,1] with the pointwise convergence topology.
Thus by Tychonoff’s Theorem the completion V˜ of V with respect to D is a compact
space, and by construction K extends continuously on this completion. Actually
µ
the completion is exactly the set of sequences (ν ) such that for each µ, K (ν )
n n 1 µ n
≥
ZIGZAG DIAGRAMS AND MARTIN BOUNDARY 5
converges, with two sequences (ν1) , (ν2) being identified whenever for each
n n 1 n n 1
µ, K (ν1) and K (ν2) have the same≥limit. ≥
µ n µ n
Denote by ∂ the set V˜ V. The latter is called the Martin entrance boundary
M
G \
of the graded graph and is a compact subset of V˜. Each element ω = lim ν in
n
G n
∂ defines a function on V by the formula →∞
M
G
ω(µ) = limK (ν ).
µ n
The following Theorem is a special case of a Theorem from Doob ([4]).
Theorem 1. With the notations above, the two following results hold:
There exists a Borel subset ∂ ∂ (called minimal boundary) such
min M
• G ⊂ G
that for any measure α harmonic with respect to Pt, there exists a unique
measure λ on ∂ giving the kernel representation
α min
G
α(µ) = K (x)dλ (x).
µ α
Z∂minG
Foranyreverserandomwalk(X ) thatrespects Pt, the path(X ,X ,...)
n n 0 0 1
• ≤ −
converges almost surely to a ∂ valued random variable X . Moreover
min
the probability that (X ) goesGth−rough µ is exactly d(µ)E(K−∞(X )).
n n 0 µ
≤ −∞
There exists a more general construction of the Martin boundary from Kunita
and Watanabe in [14], which encompasses the case of discrete random walks as well
as more general Markov processes (including the Brownian motion on a domain).
However our situation is much simpler and the previous Theorem is enough.
To summarize, the Martin entrance boundary gives a topological framework to rep-
resent harmonic measures, whereas the minimal entrance boundary gives the cor-
responding measure theoretic framework. The situation is simpler when the two
boundaries coincide. In the case of the graph that we are studying, the mini-
Z
mal entrance boundary was already described by Gnedin and Olshanski in [8]. The
purpose of the present paper is to extend this desciption to the Martin entrance
boundary by proving that the two boundary coincide.
3. The graph
Z
This section is devoted to an introduction to the graph and its relation with
Z
sequences of permutations.
3.1. Compositions. Let us first recall the definition of a composition:
Definition 1. Let n N. A composition λ of n, written λ n, is a sequence of
∈ ⊢
positive integers (λ ,...,λ ) such that λ = n.
1 r j
Let Dλ be the subset of [1;n] definedPby Dλ = {λ1,λ1 +λ2,..., 1r−1λi}. Since
there is a bijection between subsets of [1;n 1] and compositions of n, D is often
λ
− P
simply denoted by λ.
To a composition is also associated a unique ribbon Young diagram with n cells:
6 PIERRETARRAGO
each row j has λ cells, and the first cell of the row j +1 is just below the last cell
j
of the row j. For example the composition (3,2,4,1) of 10 is represented in Figure
1.
Figure 1. Skew Young tableau associated to the composition λ = (3,2,4,1).
The size n is included in the definition of composition itself, since n is equal to the
sum of all λ . If we want to emphasize the size of a compostion λ, we denote it as
j
λ . When nothing is specified, λ is always assumed to have the size n, and n always
| |
denotes the size of the composition λ.
A standard filling of a composition λ of size n is a standard filling of the associated
ribbon Young diagram: it is the assignement of an integer from 1 to n to each
cell of the composition, such that every cells have different entries, and the entries
are increasing to the right along the rows and decreasing to the bottom along the
columns. An example for the composition of Figure 1 is shown in Figure 2.
3 5 8
4 7
1 6 9 10
2
Figure 2. Standard filling of the composition (3,2,4,1).
In particular, reading the tableau from left to right and from top to bottom gives for
each standard filling a permutation σ; moreover the descent set des(σ) of σ, namely
thesetofindicesisuchthatσ(i+1) < σ(i), isexactlythesetD . Thereisabijection
λ
between the standard fillings of λ and the permutations of λ with descent set D .
λ
| |
For example the filling in Figure 2 yields the permutation (3,5,8,4,7,1,6,9,10,2).
3.2. The graph . The graded graph , which was introduced by Viennot in [22],
Z Z
is defined as follows:
(1) The set of vertices of degree n of is the set of compositions of n. The
n
Z Z
vertex of degree 0 is denoted .
∅
(2) Let λ = (λ ,...,λ ) and µ = (µ ,...,µ ) be two compositions. There is an
1 r 1 s
edge between µ and λ if and only if λ = µ +1 and
| | | |
either r = s and for each i except one µ = λ (thus exactly one µ is
• i i i0
increased by one)
ZIGZAG DIAGRAMS AND MARTIN BOUNDARY 7
either r = s + 1, and there exists j such that: for k < j, λ = µ ,
k k
•
λ + λ 1 = µ , and for k > j, λ = µ (namely one µ is split,
j j+1 j k+1 k i
−
and one cell is added at the end of the first piece).
The first four levels of are displayed in Figure 3.
Z
∅
0 1 2 3
Z Z Z Z
Figure 3. Vertices of of degree 0 to 3.
Z
For a composition λ, let Ω be the set of paths between and λ. It has been
λ
∅
shown in [22] that Ω σ S ,des(σ) = D . One way to see this is to remark
λ λ λ
≃ { ∈ | | }
that Ω is the set of all standard fillings of the ribbon diagram associated to λ. Thus
λ
these sets have same cardinality and
d(λ) = Ω = # σ S ,des(σ) = D .
λ λ λ
| | { ∈ | | }
Let P denote the uniform distribution on Ω ; from Section 2, this is equivalent to
λ λ
considering the random walk starting at λ with transition matrix Pt. This random
walk gives n random variables σλ, 1 k n, each of them being the random path
k ≤ ≤
restricted to the vertices of degree smaller than k.
Since there is a bijection between paths on from to µ and permutations of
Z ∅
µ with descent set D , each σλ is a random permutation in S , and the law of
| | µ k k
σ = σλ is the uniform distribution on the set of permutations with descent set D .
λ n λ
Moreover a counting argument yields that for σ S with des(σ) = D , under the
k µ
∈
probability P ,
λ
d(µ,λ)
(2) P (σk = σ) = .
λ λ d(λ)
By abuse of notation a finite path starting at on and the corresponding permu-
∅ Z
tation are both usually denoted by σ. In particular if σ Ω , σ denotes the path
λ k
∈
8 PIERRETARRAGO
after k steps, whereas σ(i) will denote the image of i by the permutation associated
to σ (the same for σ(A) with A a subset of 1,...,n ).
{ }
3.3. Arrangement on N. In this paragraph a permutation σ S is written as
k
∈
a word in the alphabet 1,...,k , where i = σ(j). For k 2 and σ = (i ...i ),
j 1 k
σ S is defined as {the perm}utation (i ...✓✓k...i ). If σ≥ S , σ denotes the
k 1 1 k n k
↓ ∈ − ∈ ↓
(n k) iteration of the -operation : namely all the indices between k +1 and n
− − ↓
have been erased.
An arrangement of N is a sequence (σ ,...,σ ,...) such that for all k 1, σ S ,
1 k k k
≥ ∈
and such that the following compatibility condition holds :
(σ ) = σ .
k k 1
↓ −
For example the following sequence is the first part of an arrangement :
((1),(21),(231),(2341),(52341),...).
The set of all arrangements is denoted . For k 1, let π : S be the
k k
A ≥ A →
application which consists in the projection of the sequence (σ ,σ ,...) on the k th
1 2
−
element σ . isconsidered withtheinitial topologywithrespect totheapplications
k
A
π , and with the corresponding borelian σ algebra. Thus by the Kolmogorov’s
k
−
extension Theorem, any random variable Π on is uniquely determined by the law
A
of its finite-dimensional projections (π (Π),...,π (Π)).
1 k
The result of the previous subsection yields that there is a bijection between infinite
paths on and arrangements of N, and from Section 2 this bijection extends to a
Z
bijectionbetweenharmonicmeasuresαwithrespecttoPt andrandomarrangements
Π such that
P(π (Π) = σ ,...,π (Π) = σ ) = p(des(σ )),
1 1 k k k
with p a positive function on given by p = α. This correspondance is convenient
Z d
since it allows to describe the solutions of the problem (1) in terms of random ar-
rangements.
4. Paintbox construction and Minimal boundary
Thankstothelattercorrespondance, GnedinandOlshanski describedtheminimal
entrance boundary of in terms of random arrangements. This description is the
Z
purpose of the following paragraph.
4.1. Paintbox construction. The description is based on a topological space con-
sisting in pairs of disjoint open sets of [0,1] :
Definition 2. The topological space (2) is the space
U
( (U ,U ) U and U disjoint open sets of ]0,1[ ,d),
{ ↑ ↓ | ↑ ↓ }
ZIGZAG DIAGRAMS AND MARTIN BOUNDARY 9
with the distance d between (U ,U ) and (V ,V ) given by
↑ ↓ ↑ ↓
d((U ,U ),(V ,V )) = sup(d (Uc,Vc),d (Uc,Vc)).
Haus Haus
↑ ↓ ↑ ↓ ↑ ↑ ↓ ↓
LetM ( (2))denotethesetofprobabilitymeasures withrespect totheσ algebra
1
U −
coming from the above topology.
From the definition of the metric, (U (j),U (j))) converges to (V ,V )) if and
j 1
↑ ↓ ≥ ↑ ↓
only if for each ǫ > 0:
for j large enough, the number of connected components of size larger than
•
ǫ in U and V are the same,
↑ ↑
the boundaries of the connected components of size larger than ǫ in U con-
• ↑
verge to the ones of V ,
↑
the same holds by switching and .
• ↑ ↓
In particular (U (j),U (j)) converges to ( , ) if and only if the size of the largest
↑ ↓ ∅ ∅
components in U (j) and U (j) tends to 0. The following important result holds for
(2): ↑ ↓
U
Proposition 1. (2) is compact space.
U
The minimal entrance boundary of is described by random arrangements con-
Z
structed from elements of (2).
U
Definition 3. Let U = (U ,U ) be fixed, (X ,...,X ,...) a sequence of [0,1]. For
1 k
each k 1, σ (X ,...,X )↑ ↓S is defined by the following rule:
U 1 k k
≥ ∈
(σU(X1,...,Xk))−1(i) is less than (σU(X1,...,Xk))−1(j) if and only if one of the
three following situations arises :
X and X are not in the same connected component of U or U and X < X
i j i j
• ↑ ↓
X and X are in the same connected component of U and i < j
i j
• ↑
X and X are in the same connected component of U and j < i.
i j
• ↓
The random variable σ (X ,...,X ) defined for an infinite family (X ,...,X ,...)
U 1 k 1 k
of independent uniform variables on [0,1] is denoted σ (k). The sequence
U
(σ (1),σ (2),...) is denoted σ .
U U U
The construction of σ (X ,...,X ) from (X ,...,X ) and U (2) is well-
U 1 k 1 k
∈ U
defined and unique. If U = ( , ), σ (X ,...,X ) is just the permutation as-
( , ) 1 k
∅ ∅ ∅∅
sociated to the reordering (X < X < ...X ). This permutation is denoted by
i1 i2 ik
Std 1(X ,...,X ). For each k, the random variable σ (k) has a uniform distri-
− 1 k ( , )
bution on S . ∅∅
k
The next Theorem is due to Gnedin and Olshanski in [8] (based on an important
work of Jacka and Warren in [10]) and identify (2) with the minimal entrance
U
boundary of the graded graph :
Z
Theorem 2. Each random variable σ defines a random arrangement that comes
U
A
from an harmonic probability measure on ( ,Pt), and there is an isomorphism :
Z
Φ : M ( (2)) M (∂ )
1 1 min
U −→ Z
10 PIERRETARRAGO
which restricts to a bijective map p : (2) ∂ mapping δ to σ .
U −→ minZ (U↑,U↓) (U↑,U↓)
In particular for each k 1 and σ S , P(σ (k) = σ) only depends on the
k U
≥ ∈
descent set µ of σ and is thus denoted by p (µ).
U
4.2. Martin entrance boundary of . The question is to know if ∂ = ∂ .
min M
Z Z Z
The problem is summed up in Conjecture 45 of [8]. To each composition λ of n is
associated an element U = (U (λ),U (λ)) of (2) as follows : for each s n 1 set
λ
I = [s 1, s ], and define ↑ ↓ U ≤ −
s n−1 n 1
− −
U (λ) = int( I ),U (λ) = int( I ),
s s
↑ ↓
i des(λ) i des(λ)
6∈[ ∈[
with int denoting the interior of a set. Then the conjecture states the following :
Conjecture 1. a) A sequence (λ ) is in ∂ if and only if U converges
in (2). n n≥1 MZ λn
U
b) U (U ,U ) is equivalent to K (λ ) p (µ) for all µ .
λn →U(2) ↑ ↓ µ n → (U↑,U↓) ∈ Z
c) The Martin boundary of the graph actually coincides with its minimal
Z
boundary : ∂ = (2)
M
Z U
Actually, the only difficult part is to prove the first implication of b):
(3) (U (U ,U )) = ( µ ,K (λ ) p (µ)).
λn →U(2) ↑ ↓ ⇒ ∀ ∈ Z µ n → (U↑,U↓)
Indeed suppose that the latter is true :
Proof. a) Let ω = (λ ) be in ∂ . Since (2) is compact, proving the
n n 1 M
convergence of U in≥ (2) is the Zsame as prUoving that every convergent
λn U
subsequences of U have the same limit. Let (λ ) and (λ ) be
λn φ(n) n 1 φ′(n) n 1
≥ ≥
such that
U (U1,U2),U (U2,U2)
λφ(n) → λφ′(n) →
↑ ↓ ↑ ↓
Then by (3), for all µ , ω(µ) = p (µ) and ω(µ) = p (µ). Since
U1,U1 U2,U2
∈ Z ↑ ↓ ↑ ↓
p : (2) ∂ is injective, necessarily (U1,U1) = (U2,U2). This shows
min
U → Z
that U converges. ↑ ↓ ↑ ↓
λn
Conversely if U converges in (2), the assumption (3) implies directly that
λn U
(λ ) ∂ .
n M
∈ Z
b) The direct implication is exactly (3); for the converse implication, the con-
vergence of K (λ ) for all µ implies that (λ ) ∂ . Thus from
µ n n n 1 M
a), U converges in (2). By∈inZjectivity of p, U con≥ver∈ges tZo (U ,U ).
λn U λn ↑ ↓
c) This is the summary of 1) and 2).
(cid:3)
The following sections are devoted to the proof of the implication (3), which
implies Conjecture 1 :